Bousseyroux–Potters R-transform conjecture for rotationally invariant matrix deformations

Let $\vb{A}$ be a large deterministic matrix and $\vb{B}$ a rotationally invariant random matrix. Define the matrix-valued transforms $\mathcal{G}_{\vb{A}+\vb{B}}$, $\mathcal{G}_{\vb{A}}$, and the associated scalar components $\mathfrak{g}_1=\mathfrak{g}_{1,\vb{A}+\vb{B}}$ and $\mathfrak{g}_2=\mathfrak{g}_{2,\vb{A}+\vb{B}}$. Let $\mathcal{R}_{1,\vb{B}}$ and $\mathcal{R}_{2,\vb{B}}$ denote the corresponding R-transform components. Bousseyroux–Potters R-transform conjecture. For large $\omega$ and $z\in\mathbb{C}$, one expects

$$\mathcal{G}_{\vb{A}+\vb{B}}(\omega,z)=\mathcal{G}_{\vb{A}}\left(\omega-\mathcal{R}_{1,\vb{B}}(\mathfrak{g}_1,\mathfrak{g}_2),z-\mathcal{R}_{2,\vb{B}}(\mathfrak{g}_1,\mathfrak{g}_2)\right).$$

The identity is expected to admit analytic continuation to all $(\omega,z)$, with appropriate choices of branches for $\mathcal{R}_1$ and $\mathcal{R}_2$. This is the main conjecture restated from the cited work and underlies the paper's spectral-boundary calculations; its general validity remains open.