The higher-genus open r-spin Witten conjecture

Let $r$ be the spin parameter, let $g\geq 2$, let $F_g^{x^r,o}$ denote the open $r$-spin potential, and let $\phi_g$ denote the corresponding closed $r$-spin potential. The notation $\left.\cdot\right|_{t^{r-1}_d\mapsto \frac{1}{\sqrt{-r}}(t^{r-1}_d-\delta_{d,0}rs)}$ means substituting $\frac{1}{\sqrt{-r}}(t^{r-1}_d-\delta_{d,0}rs)$ for $t^{r-1}_d$. The higher-genus open $r$-spin Witten conjecture. There exists a geometric definition of the open $r$-spin potential such that

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

This extends the genus-zero and genus-one open $r$-spin results cited in the surrounding discussion to all genera, but the higher-genus assertion is presented here as a conjectural existence statement.