Topological expansion conjecture for BGW and HCIZ integrals

Let

$$I_N^{(1)}= \int_{\operatorname{U}(N)} e^{\sqrt{z}N\operatorname{Tr}(AU+BU^{-1})}\,\mathrm{d}U,$$

and

$$I_N^{(2)}= \int_{\operatorname{U}(N)} e^{zN\operatorname{Tr}(AUBU^{-1})}\,\mathrm{d}U.$$

These are unit-mass Haar integrals depending on a complex parameter $z$ and complex matrices $A$ and $B$. Write $I_N$ for either integral and $F_N=\log I_N$.

Topological expansion conjecture. There exists a positive constant $\varepsilon$ such that, for each nonnegative integer $k$,

$$F_N=\sum_{g=0}^k N^{2-2g}F_{Ng}+o(N^{2-2k})$$

as $N\to\infty$, uniformly for complex numbers $z$ of modulus at most $\varepsilon$ and complex matrices $A,B$ of spectral radius at most $1$. The functions $F_{Ng}$ are analytic in $z$ and the eigenvalues of $A$ and $B$, with modulus uniformly bounded in $N$, and $F_{Ng}$ is a generating function for combinatorial invariants of compact connected genus $g$ Riemann surfaces.

The claim concerns the large-$N$ asymptotics of the Bars–Green/Brézin–Gross–Witten/Wadia and Harish-Chandra/Itzykson–Zuber matrix integrals, whose expansions are expected to be organized by the genus of compact connected Riemann surfaces. The supplied source describes this as a longstanding conjecture but does not provide a resolution status; the paper’s stated purpose is to prove it, so the status should be checked against the paper’s results.