The Jacobi sum-rule conjecture for the rate function

Let $\widehat{J\beta E}(N,\kappa_1N,\kappa_2N)$ denote the Jacobi beta-ensemble model with parameters $\kappa_1N$ and $\kappa_2N$, let $\nu$ be a probability measure on $[0,1]$, let $KMK$ denote the model's equilibrium measure, let ${\mathcal K}(KMK\!\ |\!\ \nu)$ denote the reversed Kullback–Leibler divergence, and let $E_j^\pm$ be the associated spectral outliers. The Jacobi sum-rule conjecture. Under the $\widehat{J\beta E}(N,\kappa_1N,\kappa_2N)$ model, the rate function is given by

$$I(\nu)={\mathcal K}(KMK\!\ |\!\ \nu)+\sum_j{\mathcal F}_J(E_j^\pm).$$

The conjecture seeks a sum-rule representation of the Jacobi large-deviation rate function in terms of equilibrium relative entropy and spectral-outlier contributions. The source says that no known sum rule was available in this general case, and the supplied text gives no evidence of a later resolution.