Self-consistent R-transform equations for rotationally invariant random matrices

Let $\mathbf{B}$ be a large rotationally invariant random matrix, with $\mathfrak{g}_{1,\mathbf{B}}$ and $\mathfrak{g}_{2,\mathbf{B}}$ denoting the corresponding transform components. For large $\omega$ and $z\in\mathbb{C}$, let $\mathcal{R}_{1,\mathbf{B}}$ and $\mathcal{R}_{2,\mathbf{B}}$ be the scalar R-transform components evaluated at $(\mathfrak{g}_1,\mathfrak{g}_2)$. Rotationally invariant R-transform conjecture. One expects

$$\mathfrak{g}_{1,\mathbf{B}}=\frac{\omega-\mathcal{R}_{1,\mathbf{B}}(\mathfrak{g}_1,\mathfrak{g}_2)}{\left(\omega-\mathcal{R}_{1,\mathbf{B}}(\mathfrak{g}_1,\mathfrak{g}_2)\right)^2-\left|z-\mathcal{R}_{2,\mathbf{B}}(\mathfrak{g}_1,\mathfrak{g}_2)\right|^2},$$

and

$$\overline{\mathfrak{g}_{2,\mathbf{B}}}=\frac{\mathcal{R}_{2,\mathbf{B}}(\mathfrak{g}_1,\mathfrak{g}_2)-z}{\left(\omega-\mathcal{R}_{1,\mathbf{B}}(\mathfrak{g}_1,\mathfrak{g}_2)\right)^2-\left|z-\mathcal{R}_{2,\mathbf{B}}(\mathfrak{g}_1,\mathfrak{g}_2)\right|^2}.$$

These identities are expected to admit analytic continuation to all $(\omega,z)$, with appropriate choices of branches of $\mathcal{R}_1$ and $\mathcal{R}_2$. This is presented as a consequence of the paper's main conjectural framework; the supplied text gives no evidence that it has been proved or disproved.