The endpoint Besov regularity conjecture for periodic symmetric-stable noises

Let $w_{\alpha}$ denote a periodic symmetric-$\alpha$-stable noise on $\mathbb{T}^d$, with $0<p<\alpha<2$. The endpoint regularity conjecture.

$$\mathbb{P}\left(w_{\alpha}\in B_{p,\infty}^{d(1/\alpha-1)}(\mathbb{T}^d)\right)=1.$$

The paper obtains regularity results below the exponent $1/\alpha-1$, while results for symmetric-stable processes on the real line motivate this endpoint assertion. Whether the stated periodic Besov regularity holds for the full range $0<p<\alpha<2$ remains open.