Smooth conjecture for enumerative integrals on spaces of maps

Fix cycles $c_i^X$ in the source $X=\mathbb{P}^k$ and cycles $c_i^Y$ in the target $Y=\mathbb{P}^n$, for $i=1,\ldots,l$. Let $\mathrm{Maps}_d(\mathbb{P}^k,\mathbb{P}^n)$ be the space of degree-$d$ holomorphic maps, and let

$$\mathrm{ev}_i\colon \mathrm{Maps}_d(\mathbb{P}^k,\mathbb{P}^n)\times \mathbb{P}^k_1\times\cdots\times\mathbb{P}^k_l\longrightarrow\mathbb{P}^n$$

be evaluation at the $i$-th source point. Denote by $\mathrm{KM}(\mathbb{P}^k,\mathbb{P}^n;\{c_i^X,c_i^Y\}|d)$ the enumerative number of maps satisfying the incidence conditions. Smooth conjecture. For any smooth closed-form representatives $\alpha_i^X\in\Omega_{cl}(\mathbb{P}^k)$ and $\alpha_i^Y\in\Omega_{cl}(\mathbb{P}^n)$ of the Poincaré dual cohomology classes of $c_i^X$ and $c_i^Y$, the two displayed integrals defining $N^{\mathrm{C,S}}$ and $N^{\mathrm{S,S}}$ are convergent and equal to $\mathrm{KM}(\mathbb{P}^k,\mathbb{P}^n;\{c_i^X,c_i^Y\}|d)$. More specifically, if the cycles have complex codimensions $n_i^X,n_i^Y$ and homology classes $d_i^{X,Y}$ times the respective generators, then the two Fubini–Study-form integrals, multiplied respectively by $\prod_i d_i^Y$ and $\prod_i(d_i^Xd_i^Y)$, both exist and equal the same enumerative number. The conjecture proposes that integrals over the space of quasimaps, smooth on the locus of actual maps, provide a numerical method for computing these enumerative invariants; the paper reports convergence experimentally and correctness in the simplest cases, while a general proof is not supplied.