Log-concavity conjecture for marginals of the sine process

Consider a determinantal point process on the real line with correlation kernel

$$K(x,y)=\frac{\sin(\pi(y-x))}{y-x}.$$

Label its points by $\cdots<Z_{-1}<Z_0<Z_1<Z_2<\cdots$ so that $Z_{-1}<0\leq Z_0$. The sine-process marginal conjecture. For every sub-collection $(Z_{i_1},\ldots,Z_{i_k})$, its marginal density on $\mathbb{R}^k$ is log-concave. The conjecture is motivated by the log-concavity of marginals for uniform Gelfand–Tsetlin functions and the identification of bead-process marginals with the sine process; the source presents it as unresolved.