The Bobkov–Madiman–Wang mixed Minkowski-sum inequality

Let $k\geq 2$, and let $A_1,A_2,\ldots,A_{k+1}$ be compact sets in $\mathbb{R}^d$. For a finite family of sets, $\sum_i A_i$ denotes their Minkowski sum, and $\operatorname{vol}$ denotes Lebesgue measure. Bobkov–Madiman–Wang conjecture.

$$\operatorname{vol}\left(\sum_{i=1}^{k+1}A_i\right)^{1/d}\geq\frac{1}{k}\sum_{i=1}^{k+1}\operatorname{vol}\left(\sum_{j\ne i}A_j\right)^{1/d}.$$

In particular, for $k=2$,

$$\operatorname{vol}(A_1+A_2+A_3)^{1/d}\geq\frac{1}{2}\left(\operatorname{vol}(A_1+A_2)^{1/d}+\operatorname{vol}(A_1+A_3)^{1/d}+\operatorname{vol}(A_2+A_3)^{1/d}\right).$$

The source describes this as a more general version of the volume monotonicity conjecture and says that star-shaped sets give a negative answer to that more general version; the displayed inequality is therefore refuted.