Spectral formula conjecture for the matrix associated with a partition

Let $\nu$ be a partition of $N$ such that the associated polynomial $P^{[\nu]}$ has only simple roots, and let $J^{[\nu]}$ be the $N\times N$ matrix defined by the paper. Let $\rho^{[\nu]}=(\rho^{[\nu]}_1,\dots,\rho^{[\nu]}_N)$ be the rational sequence associated with $\nu$.

Spectral formula conjecture. The eigenvalues of $J^{[\nu]}$ are

$$\lambda_k=2\bigl(\rho^{[\nu]}_k\bigr)^2,\qquad k=1,\dots,N,$$

not necessarily distinct. Consequently,

$$\det\bigl(J^{[\nu]}\bigr)=2^N\prod_{k=1}^N\bigl(\rho^{[\nu]}_k\bigr)^2,$$

which is a strictly positive integer.

The formula was verified numerically for all non-degenerate partitions with $N\leq 10$. For the partition $\nu=(N)$, it specializes to the Hermite case, with eigenvalues $2,2\cdot4,\dots,2N^2$.