Generalized log-local conjecture for normal crossing divisors

Let $X$ be a smooth projective variety and let $D=D_1+\cdots+D_n$ be an effective reduced normal crossing divisor with each $D_i$ smooth, irreducible, and nef. Let $I_1,\ldots,I_m$ be disjoint subsets partitioning $\{1,2,\ldots,n\}$, with $\bigcap_{i\in I_j}D_i\neq\varnothing$ for every $j$. Let $\overline{\mathcal M}_{0,0,\{(d_i)\}_{i\in I_1},\ldots,\{(d_i)\}_{i\in I_m}}(X/D,\beta)$ denote the moduli space of basic stable log maps with $m$ marked points, where the $j$th marking has maximal contact with every divisor $D_i$ for $i\in I_j$. For a curve class $\beta$, set $d_i:=D_i\cdot\beta>0$. Generalized log-local conjecture.

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

The original log-local conjecture is the special case $|I_j|=1$ for all $j$. The paper presents this as a generalization; it notes that additional interior markings may also be allowed, while no general logarithmic mirror theorem is known.