KdV tau-function conjecture for higher-valency intersection numbers

Prove that the exponential generating series of the intersection numbers $\langle\tau_{\underline d}\rangle_{m_*}$, with the variables encoding the higher-valency data $m_*$ fixed, is a tau-function for the KdV hierarchy.

Background

The paper’s proposed large-genus analysis requires precise asymptotics for intersection numbers involving Witten–Kontsevich combinatorial classes. The authors compare these numbers with ordinary tautological intersection numbers, for which recursive Virasoro relations have enabled asymptotic results.

They invoke a conjecture attributed to Kontsevich asserting a KdV tau-function structure for the relevant generating series. The paper notes that this conjecture remains unproved, although subsequent work establishes a BKP tau-function statement for a related generating function with most of the $m_*$ data fixed.