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I think there are simple counterexamples of the form $f(x,y)=\phi(x)+\psi(y)$ (here $x$ and $y$ denote real variables), where $\phi$ and $\psi$ are smooth, positive, decreasing, strictly convex functions.

The idea is that, $-\psi'(x)$ and $-\phi'(x)$ being positive decreasing functions, is not an obstruction for their ratio to oscillate between arbitrarily large and small values, on arbitrarily large intervals. This produces zig-zag orbits $z_n:=(x_n,y_n)$ for which $z_n/\|z_n\|$ may accumulate to both $(1,0)$ and $(0,1)$.

Construction. Define some smooth, positive, decreasing functions $u(t)$ and $v(t)$ such that for all $n\in\mathbb{N}$ $$u(x):={1\over (n!)^2}\quad\text{for $n$ even, and}\; n!+1\le x\le (n+2)!$$ and $$v(x):={1\over (n!)^2}\quad\text{for $n$ odd, and}\; n!+1\le x\le (n+2)!$$

(so one has only to extend them smoothly for $x\le2$ and between $n!$ and $n!+1$ to have them defined everywhere). As a consequence we have $${v(x)\over u(x)}=\begin{cases} n^{-2} &\text{for $n$ odd, and }\; n!+1\le x\le (n+1)! \\ n^2 &\text{for $n>0$ even, and }\; n!+1\le x\le (n+1)! \\ \end{cases}$$ while $$n^{-2}\le {v(x)\over u(x)}\le n^2 \quad \text{for any $n>0$ and }\; n! \le x\le n!+1 \ .$$

Note also that $u$ (respectively $v$) has finite integral on any right-unbounded interval, because the contribute to the integral on any interval $[n!+1, (n+2)!+1]$ (for $n$ even, resp. odd) is bounded by $(n+2)!/(n!)^2=(n+2)(n+1)/n!$. Therefore we can consider the smooth, decreasing, strictly convex functions $$\phi(x):=\int_x^\infty u(t)dt\quad\text{and }\quad \psi(x):=\int_x^\infty v(t)dt\; .$$

The negative gradient iteration $u_{n+1}=u_n-\eta\nabla f(u_n)$ writes, with $z_n:=(x_n,y_n)$ $$ \begin{cases} x_{n+1}=x_n+\eta \,u(x_n) \\ y_{n+1}=y_n+\eta \,v(y_n)\\ \end{cases} $$ and it follows from the above identities and inequalities on $v/u$ that it produces sequences $(x_n,y_n)$ with $\liminf_{n\to+\infty}x_n/y_n=0$ and $\limsup_{n\to+\infty}x_n/y_n=+\infty$, corresponding to a sequence $z_n/\|z_n\|$ that clusters both to $(1,0)$ and to $(0,1)$.

I think there are simple counterexamples of the form $f(x,y)=\phi(x)+\psi(y)$ (here $x$ and $y$ denote real variables), where $\phi$ and $\psi$ are smooth, positive, decreasing, strictly convex functions.

The idea is that, $-\psi'(x)$ and $-\phi'(x)$ being positive decreasing functions, is not an obstruction for their ratio to oscillate between arbitrarily large and small values, on arbitrarily large intervals. This produces zig-zag orbits $z_n:=(x_n,y_n)$ for which $z_n/\|z_n\|$ may accumulate to both $(1,0)$ and $(0,1)$.

Construction. Define some smooth, positive, decreasing functions $u(t)$ and $v(t)$ such that for all $n\in\mathbb{N}$ $$u(x):={1\over (n!)^2}\quad\text{for $n$ even, and}\; n!+1\le x\le (n+2)!$$ and $$v(x):={1\over (n!)^2}\quad\text{for $n$ odd, and}\; n!+1\le x\le (n+2)!$$

(so one has only to extend them smoothly for $x\le2$ and between $n!$ and $n!+1$ to have them defined everywhere). As a consequence we have $${v(x)\over u(x)}=\begin{cases} n^{-2} &\text{for $n$ odd, and }\; n!+1\le x\le (n+1)! \\ n^2 &\text{for $n>0$ even, and }\; n!+1\le x\le (n+1)! \\ \end{cases}$$ while $$n^{-2}\le {v(x)\over u(x)}\le n^2 \quad \text{for any $n>0$ and }\; n! \le x\le n!+1 \ .$$

Note also that $u$ (respectively $v$) has finite integral on any right-unbounded interval, because the contribute to the integral on any interval $[n!+1, (n+2)!+1]$ (for $n$ even, resp. odd) is bounded by $(n+2)!/(n!)^2=(n+2)(n+1)/n!$. Therefore we can consider the smooth, decreasing, strictly convex functions $$\phi(x):=\int_x^\infty u(t)dt\quad\text{and }\quad \psi(x):=\int_x^\infty v(t)dt\; .$$

The negative gradient iteration $u_{n+1}=u_n-\eta\nabla f(u_n)$ writes, with $z_n:=(x_n,y_n)$ $$ \begin{cases} x_{n+1}=x_n+\eta \,u(x_n) \\ y_{n+1}=y_n+\eta \,v(y_n)\\ \end{cases} $$ and it follows from the above identities and inequalities on $v/u$ that it produces sequences $(x_n,y_n)$ with $\liminf_{n\to+\infty}y_n/x_n=0$ and $\limsup_{n\to+\infty}y_n/x_n=+\infty$, corresponding to a sequence $z_n/\|z_n\|$ that clusters both to $(1,0)$ and to $(0,1)$.

Sketch of the computation. Both sequences $x_m=x_0+\eta\sum_{k=0}^{m-1}u(x_j)$ and $y_m=x_0+\eta\sum_{k=0}^{m-1}v(y_j)$ are increasing and diverging. Let's define $\mu(t)$ as the largest integer $j$ such that $x_j\le n!$ and $\nu(t)$ as the largest integer $j$ such that $y_j\le n!$. The point is to use the above identities and inequalities on $u$ and $v$ to show that, for even $n$ and $m=\mu(n!)$, the principal part in both the above sum is given by the indices $j$ such that $\mu(n!)\le j\le \mu((n+1)!)$, so that $y_m/x_m=O(1/n^2)$. Analogously, for odd $n$ and $m=\nu(n!)$, one should get $x_m/y_m=O(1/n^2)$.

I think there are simple counterexamples of the form $f(x,y)=\phi(x)+\psi(y)$ (here $x$ and $y$ denote real variables), where $\phi$ and $\psi$ are smooth, positive, decreasing, strictly convex functions. Assume that for all positive integers $n$ there holds $$|\phi'((n+1)!)|\ge n|\psi'(n!+1)|\quad \text{if $n$ is even}$$ and $$|\psi'((n+1)!)|\ge n|\phi'(n!+1)|\quad \text{if $n$ is odd}.$$ (Such a couple of convex functions can be easily realized of the form $$\phi(x):=\int_x^\infty u(t)dt\quad\text{and }\quad \psi(x):=\int_x^\infty v(t)dt\qquad $$ for suitable positive, integrable (locally at $+\infty)$), decreasing smooth functions $u$ and $v$). The negative gradient iteration $u_{n+1}=u_n-\eta\nabla f(u_n)$ writes, with $u_n:=(x_n,y_n)$ $$ \begin{cases} x_{n+1}=x_n-\eta \phi'(x_n) \\ y_{n+1}=y_n-\eta \psi'(y_n)\\ \end{cases} $$ and it follows from the above inequalities that it produces sequences $(x_n,y_n)$ with $\liminf_{n\to+\infty}x_n/y_n=0$ and $\limsup_{n\to+\infty}x_n/y_n=+\infty$, corresponding to a sequence $u_n/\|u_n\|$ that clusters both to $(1,0)$ and to $(0,1)$.

I think there are simple counterexamples of the form $f(x,y)=\phi(x)+\psi(y)$ (here $x$ and $y$ denote real variables), where $\phi$ and $\psi$ are smooth, positive, decreasing, strictly convex functions.

The idea is that, $-\psi'(x)$ and $-\phi'(x)$ being positive decreasing functions, is not an obstruction for their ratio to oscillate between arbitrarily large and small values, on arbitrarily large intervals. This produces zig-zag orbits $z_n:=(x_n,y_n)$ for which $z_n/\|z_n\|$ may accumulate to both $(1,0)$ and $(0,1)$.

Construction. Define some smooth, positive, decreasing functions $u(t)$ and $v(t)$ such that for all $n\in\mathbb{N}$ $$u(x):={1\over (n!)^2}\quad\text{for $n$ even, and}\; n!+1\le x\le (n+2)!$$ and $$v(x):={1\over (n!)^2}\quad\text{for $n$ odd, and}\; n!+1\le x\le (n+2)!$$

(so one has only to extend them smoothly for $x\le2$ and between $n!$ and $n!+1$ to have them defined everywhere). As a consequence we have $${v(x)\over u(x)}=\begin{cases} n^{-2} &\text{for $n$ odd, and }\; n!+1\le x\le (n+1)! \\ n^2 &\text{for $n>0$ even, and }\; n!+1\le x\le (n+1)! \\ \end{cases}$$ while $$n^{-2}\le {v(x)\over u(x)}\le n^2 \quad \text{for any $n>0$ and }\; n! \le x\le n!+1 \ .$$

Note also that $u$ (respectively $v$) has finite integral on any right-unbounded interval, because the contribute to the integral on any interval $[n!+1, (n+2)!+1]$ (for $n$ even, resp. odd) is bounded by $(n+2)!/(n!)^2=(n+2)(n+1)/n!$. Therefore we can consider the smooth, decreasing, strictly convex functions $$\phi(x):=\int_x^\infty u(t)dt\quad\text{and }\quad \psi(x):=\int_x^\infty v(t)dt\; .$$

The negative gradient iteration $u_{n+1}=u_n-\eta\nabla f(u_n)$ writes, with $z_n:=(x_n,y_n)$ $$ \begin{cases} x_{n+1}=x_n+\eta \,u(x_n) \\ y_{n+1}=y_n+\eta \,v(y_n)\\ \end{cases} $$ and it follows from the above identities and inequalities on $v/u$ that it produces sequences $(x_n,y_n)$ with $\liminf_{n\to+\infty}x_n/y_n=0$ and $\limsup_{n\to+\infty}x_n/y_n=+\infty$, corresponding to a sequence $z_n/\|z_n\|$ that clusters both to $(1,0)$ and to $(0,1)$.

I think there are simple counterexamples of the form $f(x,y)=\phi(x)+\psi(y)$ (here $x$ and $y$ denote real variables), where $\phi$ and $\psi$ are smooth, positive, decreasing, strictly convex functions. Assume that for all positive integers $n$ there holds $$n|\phi'((n+1)!)|\le |\psi'(n!+1)|\quad \text{if $n$ is even}$$ and $$n|\psi'((n+1)!)|\le |\phi'(n!+1)|\quad \text{if $n$ is odd}.$$ (Such a couple of convex functions can be easily realized of the form $$\phi(x):=\int_x^\infty u(t)dt\quad\text{and }\quad \psi(x):=\int_x^\infty v(t)dt\qquad $$ for suitable positive, integrable (locally at $+\infty)$), decreasing smooth functions $u$ and $v$). The negative gradient iteration $u_{n+1}=u_n-\eta\nabla f(u_n)$ writes, with $u_n:=(x_n,y_n)$ $$ \begin{cases} x_{n+1}=x_n-\eta \phi'(x_n) \\ y_{n+1}=y_n-\eta \psi'(y_n)\\ \end{cases} $$ and it follows from the above inequalities that it produces sequences $(x_n,y_n)$ with $\liminf_{n\to+\infty}x_n/y_n=0$ and $\limsup_{n\to+\infty}x_n/y_n=+\infty$, corresponding to a sequence $u_n/\|u_n\|$ that clusters both to $(1,0)$ and to $(0,1)$.

I think there are simple counterexamples of the form $f(x,y)=\phi(x)+\psi(y)$ (here $x$ and $y$ denote real variables), where $\phi$ and $\psi$ are smooth, positive, decreasing, strictly convex functions. Assume that for all positive integers $n$ there holds $$|\phi'((n+1)!)|\ge n|\psi'(n!+1)|\quad \text{if $n$ is even}$$ and $$|\psi'((n+1)!)|\ge n|\phi'(n!+1)|\quad \text{if $n$ is odd}.$$ (Such a couple of convex functions can be easily realized of the form $$\phi(x):=\int_x^\infty u(t)dt\quad\text{and }\quad \psi(x):=\int_x^\infty v(t)dt\qquad $$ for suitable positive, integrable (locally at $+\infty)$), decreasing smooth functions $u$ and $v$). The negative gradient iteration $u_{n+1}=u_n-\eta\nabla f(u_n)$ writes, with $u_n:=(x_n,y_n)$ $$ \begin{cases} x_{n+1}=x_n-\eta \phi'(x_n) \\ y_{n+1}=y_n-\eta \psi'(y_n)\\ \end{cases} $$ and it follows from the above inequalities that it produces sequences $(x_n,y_n)$ with $\liminf_{n\to+\infty}x_n/y_n=0$ and $\limsup_{n\to+\infty}x_n/y_n=+\infty$, corresponding to a sequence $u_n/\|u_n\|$ that clusters both to $(1,0)$ and to $(0,1)$.