Concavity conjecture for the fractional Dirichlet problem in dimensions at least three

Let $d\geq 3$, and let $D\subset\mathbb{R}^d$ be an arbitrary bounded convex set. Let $\varphi$ denote the solution of the fractional Dirichlet problem

$$(-\Delta)^{\alpha/2}\varphi=1\quad\text{in }D,$$

with the boundary condition from (–), for $\alpha=1$. Concavity conjecture. The solution $\varphi$ is concave on $D$. The claim would extend the planar concavity result to higher dimensions. The source explains that a generalization of a result of H. Lewy might imply it for sufficiently smooth strictly convex domains, but states that the conjecture remains open.