Any thoughts on proving the following statement, which is a generalization of the result in this question and also Theorem 1 from this paper from domain $\mathbb{R}$ to $\mathbb{R}^m$.

Suppose $f_n: \mathbb{R}^m \rightarrow (-\infty, \infty]$ is a sequence of convex functions that converges pointwise to a convex function $f: \mathbb{R}^m \rightarrow (-\infty, \infty]$ that is not monotinic in every coordinate of $\mathbb{R}^m$ , then $\inf_{x \in\mathbb{R}^m} f_n(x)$ converges to $\inf_{x\in\mathbb{R}^m} f(x)$.