Volume-ratio conjecture for linear images of the Schatten-1 ball

Let $M:\mathbb{R}^{d\times d}\to\boxed{?}$ be a linear map, and let

$$\mathcal S^d_1:=\{A\in\mathbb{R}^{d\times d}:\|A\|_*\le 1\}$$

be the Schatten-1 ball. Volume-ratio conjecture. Is it true that

$$\operatorname{vr}(M\mathcal S^d_1)\lesssim 1?$$

A positive answer would give an $O(\sqrt d)$ partial-coloring bound for the Matrix Spencer setting. The source presents this as an open problem; no resolution is supplied.