The topological-recursion tau-function conjecture for Fay identities

Let ${\cal S}$ be a spectral curve with topological-recursion invariants $F_g({\cal S})$ and $\omega_{g,n}({\cal S})$, and let $\hbar$ be a small parameter. Define the formal tau function ${\cal T}(\hbar^{-1}{\cal S})$ by the differential expansion in the theta function $\hat\Theta$, with coefficients $F_g^{(\vec m)}({\cal S})$ obtained by integrating $\omega_{g,|\vec m|}({\cal S})$ over the chosen cycles. The Fay identities are the identities

$$\frac{{\cal T}({\cal S}+D)}{{\cal T}({\cal S})}=\det\frac{{\cal T}({\cal S}+[z_j]-[\tilde z_i])}{{\cal T}({\cal S})}$$

for supersymmetric divisors $D=\sum_{i=1}^n[z_i]-[\tilde z_i]$. Tau-function conjecture from topological recursion. The formal tau function ${\cal T}(\hbar^{-1}{\cal S})$ satisfies the Fay identities, order by order in $\hbar$. This asserts that topological recursion produces a formal solution of the Fay/Hirota equations. The conjecture has been verified for the first three orders in $\hbar$ for arbitrary compact spectral curves, while the full all-orders statement is unresolved.