Conjectured log-partition function for the generalized rectangular spherical integral

Let $\bm{Y}_{\epsilon}=\bY+\sqrt{\epsilon}\,\bm{W}$ be the generalized observation matrix, with limiting noise singular-value density $\mu_\epsilon$, limiting largest singular value $\bar{\gamma}_\epsilon$, limiting largest singular value of $\bm{Y}_\epsilon$ equal to $\bar{\nu}_\epsilon$, limiting eigenvalue density of $\bY_\epsilon^\mathsf{T}\bY_\epsilon$ equal to $\hat\rho_\epsilon$, and $\bar{h}_\epsilon:=\lim_{x\downarrow\bar{\gamma}_\epsilon^2}D_{\mu_\epsilon}(x)$. Define the sticking transition by

$$\bar{\lambda}_\epsilon:=\alpha\lim_{z\downarrow\bar{\nu}_\epsilon}D_{\mu_\epsilon}(z).$$

Log-partition function conjecture. Almost surely,

$$\lim_{n\to\infty}\frac{1}{n}\ln Z_n(\bm{Y}_\epsilon)=\lim_{n\to\infty}\frac{1}{n}\ln I_n\left(\sqrt{\frac{\lambda}{\alpha}},\bm{Y}_\epsilon\right)=f_\epsilon^{(\alpha)}(\lambda,\lambda_*),$$

where $f_\epsilon^{(\alpha)}$ is the piecewise function specified in the statement, with $g_{\lambda,\epsilon}^{(\alpha)}$ as defined there; moreover, $n^{-1}\mathbb{E}\ln Z_n(\bm{Y}_\epsilon)\to f_\epsilon^{(\alpha)}(\lambda,\lambda_*)$. This conjecture gives the limiting free energy in the high- and low-temperature regimes and is used in the paper to derive the asymptotic MSE of the mismatched Bayes estimator; a rigorous proof is not supplied.