Continuity conjecture for bounded-domain nonlocal Poisson equations

Let $\Omega$ be the bounded domain in the nonlocal Poisson problem, let $u$ solve that problem, let $f\in L^2(\Omega)$, and let $\beta$ be the kernel exponent. Continuity conjecture. If

$$3<\beta<4,$$

then

$$u\in C(\Omega).$$

Here $C(\Omega)$ denotes the space of continuous functions on $\Omega$. The conjecture is motivated by regularity results for the periodic nonlocal Poisson equation and numerical observations; regularity for solutions on bounded domains is stated to remain an open problem, and no resolution is supplied.