Here's an example to show that the infimum is not always attained:

Consider the standard Hopf map $\pi:S^3\to S^2$, which is not null-homotopic, of course, so it follows that the area of the graph in $S^3\times S^2$ of any differentiable map $f:S^3\to S^2$ that is homotopic to $\pi$ is strictly greater than the area of the graph of a constant map, i.e., of $S^3\times\{c\}\subset S^3\times S^2$, where $c\in S^2$ is fixed, namely the volume of $S^3$.

Now, computation shows that, if $\phi_t:S^3\to S^3$ for $0<t\le 1$ is the $1$-parameter family of conformal maps that fix a point $p_0\in S^3$ and its antipode $-p_0\in S^3$, that is the identity for $t=1$ and expands at $p_0$ by a factor of $1/t$ for $t\in (0,1]$, then the area of the graph of $f_t = \pi\circ\phi_t$ in $S^3\times S^2$ converges to the area of $S^3\times\{\pi(p_0)\}\subset S^3\times S^2$. Indeed, the graph of $f_t$ itself converges to $S^3\times\{\pi(p_0)\}$ outside of an open neighborhood of $\{-p_0\}\times S^2$.

Consequently, the infimum of the areas of the graphs of maps $f:S^3\to S^2$ in the homotopy class of $\pi$ is the volume of $S^3$, but this infimum cannot be obtained by anything in the homotopy class.

Note: The graph of $\pi:S^3\to S^2$ in $S^3\times S^2$ is a minimal submanifold, of course, but it is not minimizing, i.e., it is unstable.

Here's an example to show that the infimum is not always attained:

Consider the standard Hopf map $\pi:S^3\to S^2$, which is not null-homotopic, of course, so it follows that the area of the graph in $S^3\times S^2$ of any differentiable map $f:S^3\to S^2$ that is homotopic to $\pi$ is strictly greater than the area of the graph of a constant map, i.e., of $S^3\times\{c\}\subset S^3\times S^2$, where $c\in S^2$ is fixed, namely the volume of $S^3$.

Now, computation shows that, if $\phi_t:S^3\to S^3$ for $0<t\le 1$ is the $1$-parameter family of conformal maps that fix a point $p_0\in S^3$ and its antipode $-p_0\in S^3$, that is the identity for $t=1$ and expands at $p_0$ by a factor of $1/t$ for $t\in (0,1]$, then the area of the graph of $f_t = \pi\circ\phi_t$ in $S^3\times S^2$ converges to the area of $S^3\times\{\pi(p_0)\}\subset S^3\times S^2$. Indeed, the graph of $f_t$ itself converges to $S^3\times\{\pi(p_0)\}$ outside of an open neighborhood of $\{-p_0\}\times S^2$.

Consequently, the infimum of the areas of the graphs of maps $f:S^3\to S^2$ in the homotopy class of $\pi$ is the volume of $S^3$, but this infimum cannot be obtained by anything in the homotopy class.

Note: The graph of $\pi:S^3\to S^2$ in $S^3\times S^2$ is a minimal submanifold, of course, but it is not minimizing, i.e., it is unstable.

Addendum: Actually, it turns out that this computation is just a special case of a much more general phenomenon. Let $\phi_t:S^{n+k}\to S^{n+k}$ for $0<t\le 1$ be the conformal dilation contracting to a fixed point $p_0\in S^{n+k}$ whose differential at $p_0$ is $t$ times the identity on $T_{p_0}S^{n+k}$. (In particular, $\phi_1$ is the identity map.) One then has the following result:

Proposition: If $f:S^{n+k}\to S^n$ is any $C^1$-map and $k>0$, then the areas of the graphs of the homotopic family $f_t = f\circ\phi_t$ in $S^{n+k}\times S^n$ converge to the volume of $S^{n+k}$ as $t$ goes to $0$.

In particular, the graph area infimum in any homotopy class in $\pi_{n+k}(S^n)$ is the same when $k>0$, namely the volume of $S^{n+k}$, and this can actually be attained only for the trivial homotopy class.

Here's an example to show that the infimum is not always attained:

Consider the standard Hopf map $\pi:S^3\to S^2$, which is not null-homotopic, of course, so it follows that the area of the graph in $S^3\times S^2$ of any differentiable map $f:S^3\to S^2$ that is homotopic to $\pi$ is strictly greater than the area of the graph of a constant map, i.e., of $S^3\times\{c\}\subset S^3\times S^2$, where $c\in S^2$ is fixed, namely the volume of $S^3$.

Now, computation shows that, if $\phi_t:S^3\to S^3$ for $0<t\le 1$ is the $1$-parameter family of conformal maps that fix a point $p_0\in S^3$ and its antipode $-p_0\in S^3$, that is the identity for $t=1$ and expands at $p_0$ by a factor of $1/t$ for $t\in (0,1]$, then the area of the graph of $f_t = \pi\circ\phi_t$ in $S^3\times S^2$ converges to the area of $S^3\times\{\pi(p_0)\}\subset S^3\times S^2$. Indeed, the graph of $f_t$ itself converges to $S^3\times\{\pi(p_0)\}$ outside of an open neighborhood of $\{-p_0\}\times S^2$.

Consequently, the infimum of the areas of the graphs of maps $f:S^3\to S^2$ in the homotopy class of $\pi$ is the volume of $S^3$, but this infimum cannot be obtained by anything in the homotopy class.

Here's an example to show that the infimum is not always attained:

Consider the standard Hopf map $\pi:S^3\to S^2$, which is not null-homotopic, of course, so it follows that the area of the graph in $S^3\times S^2$ of any differentiable map $f:S^3\to S^2$ that is homotopic to $\pi$ is strictly greater than the area of the graph of a constant map, i.e., of $S^3\times\{c\}\subset S^3\times S^2$, where $c\in S^2$ is fixed, namely the volume of $S^3$.

Now, computation shows that, if $\phi_t:S^3\to S^3$ for $0<t\le 1$ is the $1$-parameter family of conformal maps that fix a point $p_0\in S^3$ and its antipode $-p_0\in S^3$, that is the identity for $t=1$ and expands at $p_0$ by a factor of $1/t$ for $t\in (0,1]$, then the area of the graph of $f_t = \pi\circ\phi_t$ in $S^3\times S^2$ converges to the area of $S^3\times\{\pi(p_0)\}\subset S^3\times S^2$. Indeed, the graph of $f_t$ itself converges to $S^3\times\{\pi(p_0)\}$ outside of an open neighborhood of $\{-p_0\}\times S^2$.

Consequently, the infimum of the areas of the graphs of maps $f:S^3\to S^2$ in the homotopy class of $\pi$ is the volume of $S^3$, but this infimum cannot be obtained by anything in the homotopy class.

Note: The graph of $\pi:S^3\to S^2$ in $S^3\times S^2$ is a minimal submanifold, of course, but it is not minimizing, i.e., it is unstable.