Convex non-squeezing conjecture in intermediate dimensions

Let $1<k<m$, let $\varphi\in\operatorname{Symp}(\mathbb R^{2m})$, and let $P_{\mathbb R^{2k}}:\mathbb R^{2m}\to\mathbb R^{2k}$ be projection onto the first $2k$ coordinates. Assume that $\varphi(B^{2m})$ is convex. Convex intermediate-dimensional non-squeezing conjecture.

$$\operatorname{vol}_{\mathbb R^{2k}}\bigl(P_{\mathbb R^{2k}}(\varphi(B^{2m}))\bigr)\geq\frac{\pi^k}{k!}.$$

The unrestricted statement is false for $1<k<m$, but the source proposes that convexity of the symplectic image restores the lower bound; no resolution is given.