Fractional torsion-function concavity conjectures

Let $d\geq 2$, let $D\subset\mathbb{R}^d$ be an arbitrary bounded convex set, and let $\varphi$ be the solution of the fractional Dirichlet problem

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

with the boundary condition from (–). Fractional concavity conjectures. (a) If $\alpha\in(1,2)$, then $\varphi$ is $1/\alpha$-concave on $D$. (b) If $\alpha\in(0,1)$, then $\varphi$ is concave on $D$. These conjectures extend the established concavity phenomenon for the half-Laplacian to other fractional orders and arbitrary bounded convex domains. The supplied text does not state a resolution.