Let $f$ be a function of $\geq 2$ real variables defined on a convex cone $\mathcal{C}$ in the upper half plane, with $f(0) = 0$. Suppose $f$ is subadditive, i.e. $f(x_1+y_1, \dots, x_n+y_n) \leq f(x_1, \dots, x_n) + f(y_1, \dots, y_n)$ in its domain, $f \geq 0$, and $f$ is nondecreasing in $x$ and nonincreasing in $y$.

I'd like to know whether $f$ is necessarily $\left(0,1\right]$-superhomogeneous, i.e., whether $f(\lambda x) \geq \lambda f(x)$ for $\lambda \in \left(0,1\right]$.

I can show that $f$ is superhomogeneous for $\lambda$ as above in $\mathbf{N}^{-1}$, I'd like to know if that's true on the interval $\left(0,1\right]$.