Wishart-prior matrix generalized linear model conjecture

Let $\Phi_d^{(G)}$ be the free entropy of the Gaussian-equivalent matrix generalized linear model, with $\bS^\star\sim\mcW_{m,d}$, and let $Q_0$, $\alpha$, $J_q$, and $\Psi$ have the meanings in the model. Define

$$\Psi(\hq)\coloneqq\frac14+\lim_{d\to\infty}\frac1{d^2}\mathbb E_{\bY}\log\mathbb E_{\bS\sim\mcW_{m,d}}\exp\left(-\frac d4\operatorname{Tr}[(\bY-\sqrt{\hq}\bS)^2]\right),$$

where $\bY=\sqrt{\hq}\bS^\star+\bxi$, $\bxi\sim\operatorname{GOE}(d)$, and $\bS^\star\sim\mcW_{m,d}$. The free entropy of a matrix generalized linear model. Assuming the limit defining $\Psi$ is well-defined,

$$\lim_{d\to\infty}\Phi_d^{(G)}=\sup_{q\in[1,Q_0]}\inf_{\hq\geq0}\left[\frac{(Q_0-q)\hq}{4}+\Psi(\hq)+\alpha\int_{\mathbb R\times\mathbb R}\mathrm dy\,\mcD\xi\,J_q(y,\xi)\log J_q(y,\xi)\right].$$

This extends the generalized-linear-model free-entropy formula to the dependent Wishart prior; the source identifies technical difficulties in establishing it rigorously, so it remains conjectural.