The sine-process hereditary completeness conjecture

Let $X$ be a configuration sampled from the sine process, with law denoted by ${{\mathbb P}_{\mathscr S}}$, and let $p\in X$ be a particle. A set is hereditarily complete if every bounded measurable partition of it yields a complete system of reproducing kernels in the associated Paley–Wiener space. The sine-process hereditary completeness conjecture. For ${{\mathbb P}_{\mathscr S}}$-almost every configuration $X$ and any particle $p\in X$, the set $X\setminus p$ is hereditarily complete. The paper's main theorem establishes uniqueness after removing one particle, but the stronger hereditary-completeness assertion is not resolved in the supplied text.