Uniqueness conjecture for Pólya ensembles on Hermitian matrix spaces

Let $H_m^{(n)}$ denote the relevant space of Hermitian matrices, and let a polynomial ensemble on $H_m^{(n)}$ have weights

$$w_b(s_c)=(-s_c\partial_c)^{b-1}\omega(s_c)$$

as in Eq. (pol.H). A random matrix $g$ is drawn from a Pólya ensemble on $G_{m,n}^{(n)}$, and the induced Hermitian matrix is $x=\pm gg^*$. Pólya-ensemble uniqueness conjecture. The only polynomial ensembles on $H_m^{(n)}$ whose weights have the displayed form are those induced by $x=\pm gg^*$ with $g$ drawn from a Pólya ensemble on $G_{m,n}^{(n)}$. This conjecture asserts that these induced ensembles exhaust the realizations compatible with the stated polynomial-ensemble structure; the source presents it because a proof is lacking and does not provide a resolution.