The threshold conjecture for concavity of stable distributions

A strictly stable probability measure is a stable law, and a probability measure is convex when it satisfies the paper's convexity property for measures. In finite dimensions, fix $\kappa\leq 0$ and let $\alpha^*(\kappa,n)\in(0,2]$ depend only on $\kappa$ and $n$. Threshold conjecture for concavity of stable distributions. Any strictly stable probability measure on an infinite-dimensional separable Hilbert space is convex. In the finite-dimensional case, a spherically symmetric stable distribution of index $\alpha$ on $\mathbb{R}^n$ is $\kappa$-concave if and only if

$$\alpha\geq\alpha^*(\kappa,n).$$

The conjecture predicts both an infinite-dimensional convexity phenomenon and a sharp threshold in finite dimensions. The surrounding discussion gives partial negative information for symmetric stable laws but does not establish the claimed threshold.