Rationality of descendent series for Hilbert and Quot schemes of surfaces

Quot schemes of quotients a trivial bundle arbitrary rank on nonsingular projective surface X carry perfect obstruction theories and virtual fundamental classes whenever the quotient sheaf has at most 1-dimensional support. The associated generating series Euler characteristics was conjectured to be rational function in Oprea Pandharipande (in, Geom Topol. http://arxiv.org/abs/1903.08787 ) when is simply connected. We conjecture here rationality more general descendent with insertions obtained from Chern characters tautological sheaf. prove Hilbert scheme cases for all curve class 0.