Ball-support conjecture for simple fractional-Laplacian eigenvalue minimizers

Let $k\geq 2$, let $A$ be a $\lambda_k$-minimizer, and let $u_k$ be the corresponding eigenfunction. Assume that $\lambda_k$ is simple. Ball-support conjecture. For every copy ${\mathbb R}_i^n$ of ${\mathbb R}^n$, the set

$$\{u_k|_{{\mathbb R}_i^n}\neq 0\}$$

is a ball. The preceding discussion expects minimizing configurations to be unions of balls in different copies of ${\mathbb R}^n$. The conjecture is presented as an expected structural property under simplicity; the paper notes special verification when the domain consists of two or three disjoint connected sets, but does not resolve the general case.