Witten's generalized conjecture for higher-spin curves

Let $r$-spin curves have moduli spaces $\overline{\mathcal{M}}_{g,n}^{1/r,\mathbf{m}}$, and let $\Psi(\mathbf{t})$ be the potential of the $\mathrm{KdV}_r$ hierarchy. Intersection numbers on these spaces are intended to define this potential. Witten's generalized conjecture. There exists an $r$-spin virtual cohomology class $c^{1/r}$ on $\overline{\mathcal{M}}_{g,n}^{1/r,\mathbf{m}}$ satisfying Axioms 1–5 of the relevant definition, such that the large phase space potential $\Phi(\mathbf{t})$ of the $r$-spin cohomological field theory coincides with $\Psi(\mathbf{t})$. The conjecture refines Witten's original formulation by requiring the factorization properties encoded in the CohFT axioms; the paper notes genus-zero and genus-one evidence, but the full statement is not established here.