Godberson's conjecture on mixed volumes of a convex body and its reflection

Let $K\subseteq\mathbb R^n$ be a convex body. For $0<j<n$, write $V(K[j],-K[n-j])$ for the mixed volume with $K$ repeated $j$ times and $-K$ repeated $n-j$ times, and let $\operatorname{Vol}(K)$ denote its volume.

Godberson's conjecture.

$$V(K[j],-K[n-j])\leq \binom{n}{j}\operatorname{Vol}(K).$$

Equality holds if and only if $K$ is a simplex.

This conjecture would strengthen the Rogers–Shephard inequality for difference bodies term by term. The supplied text does not indicate whether it has been resolved.