Unbalanced difference-body conjecture for simplices

Let $K\subset\mathbb{R}^n$ be a convex body, let $\lambda\in(0,1)$, and define the unbalanced difference body

$$D_\lambda K=(1-\lambda)K+\lambda(-K).$$

Let $\Delta$ be an $n$-dimensional simplex. Unbalanced difference-body conjecture. One has

$$\frac{\operatorname{Vol}(D_\lambda K)}{\operatorname{Vol}(K)}\leq \frac{\operatorname{Vol}(D_\lambda\Delta)}{\operatorname{Vol}(\Delta)},$$

with equality if and only if $K$ is an $n$-simplex.

At $\lambda=\frac12$, this specializes, up to scaling, to the Rogers–Shephard difference-body setting. The conjecture would follow from Godbersen's conjecture, but the stated assertion is not established in the source and remains open.