Generalized local-orbifold conjecture for normal crossing divisors

Let $X$ be a smooth projective variety, let $D=D_1+\cdots+D_n$ be an effective reduced normal crossing divisor with smooth, irreducible, nef components, and let $I_1,\ldots,I_m$ partition $\{1,\ldots,n\}$ with $\bigcap_{i\in I_j}D_i\neq\varnothing$. Let $X_{D,\vec r}$ be the associated multi-root stack, and let $F$ denote the morphism from the relevant orbifold stable-map moduli space to the moduli space of stable maps to $\bigoplus_{i=1}^n\mathcal O_X(-D_i)$. For a curve class $\beta$, set $d_i:=D_i\cdot\beta>0$. Generalized local-orbifold conjecture. When the $r_i$ are sufficiently large,

$$\left[\left(\bigcup_{j=1}^m\operatorname{ev}_j^*\left(\bigcup_{i\in I_j}D_i\right)\right)\cap\overline{\mathcal M}_{0,m}\left(\bigoplus_{i=1}^n\mathcal O_X(-D_i),\beta\right)\right]^{\operatorname{vir}} =\left(\prod_{i=1}^n(-1)^{d_i-1}\right)F_*[\overline{\mathcal M}_{0,0,\{(d_i)\}_{i\in I_1},\ldots,\{(d_i)\}_{i\in I_m}}(X_{D,\vec r},\beta)]^{\operatorname{vir}}.$$

This conjecture relates local invariants to orbifold invariants of multi-root stacks. The paper proposes it because the relevant local mirror theorem is known, whereas a mirror theorem for logarithmic Gromov–Witten invariants is not; the large-root condition is essential.