Conjecture on extending the eigenvalue formula for $L_{n,p,k}$ to all $p$ and $n$

Let $L_{n,p,k}$ be the matrix whose scaled eigenvalues and grids are described by Theorem~, with $1\le p\le100$, $0\le k\le\min(1,p-1)$, and the grids $\Theta_{n,p,k}^{\langle j\rangle}$ and functions $e_{p,k}$ as in that theorem. The eigenvalue-extension conjecture. Theorem~ continues to hold when “$1\le p\le100$” is replaced by “$p\ge1$” and “for every $n=1,\ldots,20$” is replaced by “for every $n\ge1$”. This conjecturally extends the finite-range formula for the eigenvalues of $n^{-2}L_{n,p,k}$ to every positive $p$ and every positive matrix size $n$; the source gives no resolution.