BNPT geometric-definition conjecture for open Gromov–Witten invariants

Let $S$ be a moduli specification, let $\vec{a}$ be a vector of nonnegative integers, and let $\vec{\gamma}$ be a vector of cohomology classes in $H^*_T(\mathbb{P}^1)$. The combinatorial open Gromov–Witten invariant is denoted by $\operatorname{OGW}(S,\vec{a},\vec{\gamma})$. BNPT geometric-definition conjecture. There exists a geometric definition for $\operatorname{OGW}(S,\vec{a},\vec{\gamma})$ for all moduli specifications $S$. The conjecture proposes a geometric realization of the combinatorial formula via coherent boundary conditions; it is proved in genus $0$, while the all-genus assertion remains open.