The Laguerre sum-rule conjecture for the rate function

Let $\nu$ be a probability measure on $(0,\infty)$, let $MP$ denote the Marchenko–Pastur measure, let ${\mathcal K}(MP\!\ |\!\ \nu)$ denote the reversed Kullback–Leibler divergence, and let $E_j^\pm$ be the spectral outliers associated with the Jacobi parameters of $\nu$. The Laguerre sum-rule conjecture. The rate function is

$$I^{\tt w}(\nu)={\mathcal K}(MP\!\ |\!\ \nu)+\sum_j{\mathcal F}_L(E_j^\pm).$$

This proposes a sum-rule expression for the large-deviation rate function in terms of the reversed relative entropy with respect to the Marchenko–Pastur law and contributions from spectral outliers; the supplied text does not state whether it has been proved or refuted.