Godbersen's conjecture for mixed volumes of a convex body and its reflection

Let $K\subset\mathbb{R}^n$ be a convex body, and let $V$ denote mixed volume, with $K[k]$ denoting $k$ copies of $K$. Godbersen's conjecture. For each convex body $K\subset\mathbb{R}^n$ and each $0\leq k\leq n$ one has

$$V(-K[k],K[n-k])\leq {n\choose k}\operatorname{vol}_n(K).$$

Moreover, if $0<k<n$ and $K$ has nonempty interior, equality holds if and only if $K$ is a simplex. This conjecture refines the Rogers--Shephard inequality term by term; it is known in several special cases, including anti-blocking convex bodies, but is not resolved in general.