ELSV formula for leaky completed-cycle Hurwitz numbers

Establish an ELSV-type formula for single connected completed-cycle leaky Hurwitz numbers, thereby obtaining a leaky analogue of the Zvonkine conjecture.

Background

The paper develops (r+1)-completed-cycle k-leaky double Hurwitz numbers through the semi-infinite wedge formalism and proves their piecewise polynomiality, wall-crossing formulae, and cut-and-join equations. An ELSV-type formula would provide a geometric interpretation of the single connected invariants as integrals over moduli spaces of curves, analogous to known ELSV formulae for several other Hurwitz theories.

The authors relate this problem to the Zvonkine conjecture, which has since been proved in the non-leaky setting, and note that k-leaky Hurwitz numbers with descendants were originally defined using an ELSV-type intersection-theoretic construction. The unresolved task is to identify the corresponding formula for completed-cycle leaky Hurwitz numbers without descendants.