The Bobkov–Madiman–Wang volume monotonicity conjecture for Minkowski sums

Let $A$ be a compact set in $d^d$ for some $d\in\mathbb{N}$. For $k\in\mathbb{N}$, write

$$A[k]=\sum_{i=1}^k A.$$

The volume is the Lebesgue measure, denoted by $\operatorname{vol}$. Bobkov–Madiman–Wang conjecture. The sequence

$$\left\{\operatorname{vol}\left(\frac{1}{k}A[k]\right)\right\}_{k\geq 1}$$

is non-decreasing in $k$. This conjecture asks whether the volume convergence of $\frac{1}{k}A[k]$ to the convex hull of $A$ is monotone; it is a proposed extension of the evident equality for convex sets and the Shapley–Folkman–Starr convergence theorem. The supplied text gives no general resolution, so the conjecture is recorded as open.