A conjectured local Liakopoulos–Meyer inequality for convex bodies

Let $K$ be a convex body in $\mathbb{R}^n$ and let $(\sigma_1,\dots,\sigma_m)$ be an $s$-cover of $\sigma\subset[n]$. For each $\tau\subset[n]$, let $H_\tau$ denote the coordinate subspace associated with $\tau$, and let $|\cdot|$ denote volume. Local Liakopoulos–Meyer conjecture. Then

$$|K|^{m-s}\max_{x\in\mathbb{R}^n}|K\cap(x+H_\sigma^\bot)|^s\geq \frac{\prod_{j=1}^m(|\sigma|-|\sigma_j|)!}{|\sigma|!^{m-s}}\prod_{j=1}^m|K\cap H_{\sigma_j}^\bot|.$$

This is presented as a conjectural strengthening or formulation of the preceding inequalities for arbitrary convex bodies and $s$-covers; the supplied text does not indicate whether it has been proved or remains open.