The rGD wave function conjecture of BCT3

Let $r\geq 2$, let $L$ be the Gelfand–Dikii operator, and let $\Phi(T_*,\lambda)$ solve

$$\frac{\partial\Phi}{\partial T_n}=\lambda^{n-1}(L^{n/r})_+\Phi\quad(n\geq 1),\qquad \Phi\big|_{T_{\geq 2}=0}=1.$$

Write $\phi=\log\Phi=\sum_{g\in\mathbb Z}\lambda^{g-1}\phi_g$, and let $F_g^{\frac{1}{r},o}(t^*_*,s)$ be the genus-$g$ open $r$-spin potential formed from the open intersection numbers $\langle\tau^{\alpha_1}_{d_1}\cdots\tau^{\alpha_l}_{d_l}\sigma^k\rangle_g^{\frac{1}{r},o}$. The rGD wave function conjecture of BCT3. For every genus $g\geq 1$, there is a geometric construction of these open $r$-spin intersection numbers, and

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

This proposes an all-genus geometric open $r$-spin theory and identifies its potentials with coefficients of the logarithm of the $r$GD wave function; the genus-zero analogue is described earlier, while the higher-genus geometric construction remains conjectural.