The closed extended r-spin potential conjecture for the Gelfand–Dickey wave function

Assume that for every $b\geq 1$, $g,n\geq 0$, $0\leq\alpha_1,\ldots,\alpha_n\leq r-1$, and $d_1,\ldots,d_n\geq 0$, the closed extended correlators $\left\langle\tau^{\alpha_1}_{d_1}\cdots\tau^{\alpha_n}_{d_n}(\tau^{-1}_0)^b\right\rangle^{\frac{1}{r},\mathrm{ext}}_g$ have been geometrically constructed. Let $F_g^{\frac{1}{r},\mathrm{ext}}(t^*_*)$ be the generating series defined from these correlators, and let $\phi_g$ denote the genus-$g$ part of the Gelfand–Dickey wave function. Write $t^{\leq r-2}_*$ for the variables with superscripts at most $r-2$ and $t^{r-1}_*$ for those with superscript $r-1$.

Closed extended r-spin potential conjecture. For any $g\geq 0$,

$$F^{\frac{1}{r},\mathrm{ext}}_g=(-r)^{\frac{1-g}{2}}\phi_g\left(t^{\leq r-2}_*,\frac{1}{\sqrt{-r}}t^{r-1}_*\right).$$

This identifies the closed extended $r$-spin generating series with a rescaled special solution of the Gelfand–Dickey hierarchy; because the correlators themselves are conjectural beyond the established cases, the source leaves this claim unresolved.