The extended bi-graded Toda conjecture for orbifold descendent potentials

Let $\mathcal{C}_{k,m}$ be the orbifold obtained from $\mathbb{P}^1$ by replacing neighborhoods of $0$ and $\infty$ with the orbifold discs $D_1/\mathbb{Z}_k$ and $D_2/\mathbb{Z}_m$, and let $M_{k,m}$ be the corresponding Frobenius manifold. Let $\mathcal{D}_{\mathcal{C}_{k,m}}$ denote the total descendent potential of $\mathcal{C}_{k,m}$. Extended bi-graded Toda conjecture. The total descendent potential $\mathcal{D}_{\mathcal{C}_{k,m}}$ is a tau-function of the extended bi-graded Toda hierarchy corresponding to $M_{k,m}$. This conjecture connects the orbifold Gromov–Witten theory of $\mathcal{C}_{k,m}$ with the integrable hierarchy introduced by G. Carlet. The supplied text gives no resolution; the preceding paragraph notes that the relevant Givental formula can be proved in this case by virtual localization.