The higher-genus open $r$-spin potential formula

Let $r\ge 2$, let $L$ be the specified solution of the $r$-th Gelfand--Dickey hierarchy, and let $\Phi$ solve

$$\frac{\partial\Phi}{\partial T_n}=\varepsilon^{n-1}(L^{n/r})_+\Phi,$$

with $\Phi|_{T_{\ge 2}=0}=1$. Write $\phi=\log\Phi=\sum_{g\in\mathbb{Z}}\varepsilon^{g-1}\phi_g$, and identify the variables $T_{mr}$ with $t^{r-1}_{m-1}$ as specified in the source. Higher-genus open $r$-spin potential conjecture. For any $g\ge 1$,

$$\mathcal{F}^{\frac{1}{r},o}_g=\left.(-r)^{\frac{g-1}{2}}\phi_g\right|_{t^{r-1}_d\mapsto\frac{1}{\sqrt{-r}}(t^{r-1}_d-\delta_{d,0}rs)}.$$

If the conjectural higher-genus open intersection numbers exist, this formula would express their generating potentials through the higher-genus components of the hierarchy solution; the paper supplies the genus-zero analogue but not a higher-genus geometric definition.