The multivariable random unitary pencil determinant limit conjecture

For a $g$-tuple of matrices $\mathcal X=(X_1,\dots,X_g)$, define its outer spectral radius by

$$\textbf{rad}(\mathcal X)=\operatorname{rad}\left(\sum_{j=1}^g X_j\otimes\overline{X_j}\right)^{1/2},$$

where $\operatorname{rad}(X)$ is the ordinary spectral radius. Let $\mathcal U=(U_1,\dots,U_g)$ range over $U(d)^g$, and let $L_{\mathcal X}(\mathcal U)$ denote the associated linear pencil. The multivariable determinant limit conjecture. If $\mathcal X$ and $\mathcal Y$ are $g$-tuples of square matrices of sizes $k\times k$ and $k'\times k'$, respectively, with $\textbf{rad}(\mathcal X),\textbf{rad}(\mathcal Y)<1$, then

$$\lim_{d\to\infty}\int_{U(d)^g}\det(L_{\mathcal X}(\mathcal U))\overline{\det(L_{\mathcal Y}(\mathcal U))}\,d\mathcal U =\det\left(I_k\otimes I_{k'}-\sum_{j=1}^gX_j\otimes\overline{Y_j}\right)^{-1}.$$

The conjecture extends the one-variable determinant limit to several independent Haar-unitary variables and proposes that the outer spectral-radius ball is the correct domain of convergence. The diagonal and triangular cases provide supporting results, but the general matrix-coefficient case remains open.