Eventual concavity conjecture for parallel volume

Let $A$ be a compact subset of $\mathbb{R}^n$, let $B$ be a convex body in $\mathbb{R}^n$, and define

$$V_{A,B}(t)=|A+tB|.$$

Eventual concavity conjecture. There exists $t_0$ such that $V_{A,B}(t)$ is $\frac{1}{n}$-concave on $[t_0,+\infty)$. This is proposed as a weaker form of the Costa–Cover conjecture after the latter is shown to fail in general. The paper establishes the claim only for special sets, so the general assertion remains open.