Genus-like expansion conjecture for D2- and M2-brane path integrals

Let $Y$ be a target space-time, and let $Z_{D2}(Y)$ and $Z_{M2}(Y)$ denote the path integrals of perturbative D2-brane and M2-brane theories, respectively. The relevant worldvolumes are $3$-manifolds obtained from connected sums of finite disjoint unions of copies of $S^2\times S^1$. Genus-like expansion conjecture. There exists a built-in natural genus-like expansion for the path integral $Z_{D2}(Y)$ of a perturbative D2-brane theory. Similarly, there exists such an expansion for the path integral $Z_{M2}(Y)$ of a perturbative M2-brane theory. This is suggested by the analogous reasoning for D3-branes and by the role of branched covers and connected sums of $S^2\times S^1$ in the D2/M2-brane setting; the source does not state a resolution.