Recht-Ré conjecture with norms inside the sums

Let $A_1,\dots,A_n$ be positive semidefinite matrices, and let $\lVert\,\cdot\,\rVert$ denote the matrix norm used in the conjecture. For ordered products of length $m$, distinguish sums over all index tuples from sums over tuples with distinct indices. Norm-sum Recht-Ré conjecture.

$$\frac{1}{n^m}\sum_{1\leq j_1,\dots,j_m\leq n}\lVert A_{j_1}\cdots A_{j_m}\rVert\geq\frac{(n-m)!}{n!}\sum_{\substack{1\leq j_1,\dots,j_m\leq n,\j_1,\dots,j_m\ \text{distinct}}}\lVert A_{j_1}\cdots A_{j_m}\rVert.$$

This is a related conjecture attributed to Recht and Ré, with the norms taken termwise rather than after summation. The source does not state whether it has been resolved.