The replica conjecture for the non-Hermitian R-transform of a unitary sum

Let $\mathbf{A}$ and $\mathbf{B}$ be two large independent random matrices, and let $\mathbf{U}$ be a Haar-distributed unitary random matrix. For large $z\in\mathbb{C}$ or large $\omega\in\mathbb{R}$, let $\mathcal{G}_{\mathbf{A}}(\omega,z)$ denote the matrix-valued transform used in the paper, and let $\mathfrak{g}_1(\omega,z)$ and $\mathfrak{g}_2(\omega,z)$ be its scalar components. Replica conjecture. One expects

$$\mathbb{E}_{\mathbf{U}}\!\left[\mathcal{G}_{\mathbf{A}+\mathbf{U}\mathbf{B}\mathbf{U}^*}(\omega,z)\right] = \mathcal{G}_{\mathbf{A}}\!\left(\omega-\mathcal{R}_{1,\mathbf{B}}(\mathfrak{g}_1(\omega,z),\mathfrak{g}_2(\omega,z)),\,z-\mathcal{R}_{2,\mathbf{B}}(\mathfrak{g}_1(\omega,z),\mathfrak{g}_2(\omega,z))\right).$$

This identity is expected to admit an analytic continuation to all $(\omega,z)$, with appropriate choices of branches of the multivalued extensions $\widetilde{\mathcal{R}_1}$ and $\widetilde{\mathcal{R}_2}$. The conjecture is the paper's main replica-method prediction for addition of large independent non-Hermitian random matrices under unitary conjugation; the supplied text gives no evidence of resolution.