Positivity conjecture for the fractional Bakry–Émery curvature constant

Let $\kappa(\gamma)$ denote the limiting curvature constant associated with the fractional Laplacian parameter $\gamma$, as defined in the surrounding results. Positivity conjecture. For all $\gamma \in (1,2)$,

$$\kappa(\gamma)>0.$$

The claim concerns the unresolved positivity of the limiting constant in the supercritical range $1<\gamma<2$; the supplied context establishes global optimality of $\gamma=1$ but does not resolve this positivity assertion.