Rationality of cohomological descendent series for Quot schemes on surfaces with $p_g=0$

For a smooth projective surface $S$, Johnson--Oprea--Pandharipande defined cohomological descendent generating series for Quot schemes of rank-$0$ quotients of $\OO_S^{\oplus N}$. We prove rationality of these series in the remaining cohomological surface case \[ p_g(S)=0,\qquad β\neq 0,\qquad N>1.