Concavity conjecture for the ground state of the one-dimensional symmetric stable process

Let $I=(-1,1)$, and let $\varphi_1^\alpha$ be the ground state eigenfunction for the symmetric stable process of index $0<\alpha<2$ killed upon leaving $I$. Concavity conjecture. The function $\varphi_1^\alpha$ is concave on $I$. The only known case is $\alpha=1$, where concavity was proved for the Cauchy process; the conjecture proposes this property for all symmetric stable processes in the interval.