The Dubrovin–Yang conjecture on the special cubic Hodge hierarchy

Let $p,q,r e 0$ satisfy the local Calabi–Yau condition $1/p+1/q+1/r=0$, with $r=-pq/(p+q)$. Set $s_j=-(2j-2)!(p^{2j-1}+q^{2j-1}+r^{2j-1})$, $alpha=p/\sqrt{p+q}$, and $beta=q/\sqrt{p+q}$. Let $\mathcal{H}(\mathbf{t},\mathbf{s};\varepsilon)$ be the corresponding free energy, and define $u$ and the variables $x_k,y_k$ by the formulas in the source. Let the fractional Volterra hierarchy be the integrable hierarchy defined by its flows in the paper. Dubrovin–Yang conjecture. The recombined Hodge hierarchy coincides with the fractional Volterra hierarchy after the rescaling $x_k=\alpha k\binom{((\alpha+\beta)/\beta)k}{k}t^{1,k}$ and $y_k=\beta k\binom{((\alpha+\beta)/\alpha)k}{k}t^{2,k}$. This conjecture proposes an integrable-system description of special cubic Hodge integrals; the source provides no resolution evidence.