The genus zero log/local/open correspondence for quasi-tame Looijenga pairs

A Looijenga pair $Y(D)=(Y,D)$ consists of a smooth rational complex projective surface $Y$ and an anticanonical simple normal crossings divisor $D=\bigcup_{i=1}^lD_i$, with each $D_i$ smooth, irreducible, and nef. Write

$$E_{Y(D)}=\operatorname{Tot}\left(\bigoplus_i\mathcal{O}_Y(-D_i)\right).$$

A nef pair is quasi-tame if $E_{Y(D)}$ is deformation-equivalent to the total space associated to a tame Looijenga pair. Let $Y^{\rm op}(D)$ denote the corresponding open geometry, let $d\in H_2(Y(D),\mathbb{Z})$ be a curve class, and let $N_d(E_{Y(D)})$, $O_{0,d}(Y^{\rm op}(D))$, and $N_{0,d}^{\rm \log}(Y(D))$ denote the genus-zero local, open, and logarithmic Gromov--Witten invariants. Let $\mathrm{GV}_d$, $\mathrm{LMOV}_{0,d}$, and $\mathrm{DT}^{\rm num}_d$ denote the associated BPS and numerical Donaldson--Thomas invariants, and let $\mathsf{Q}(Y(D))$ be the associated quiver.

The genus zero log/local/open correspondence. The invariants satisfy

$$N_d(E_{Y(D)})=O_{0,d}(Y^{\rm op}(D))=\left(\prod_{j=1}^l\frac{(-1)^{d\cdot D_j-1}}{d\cdot D_j}\right)N_{0,d}^{\rm \log}(Y(D)),$$

and

$$\mathrm{GV}_d(E_{Y(D)})=\mathrm{LMOV}_{0,d}(Y^{\rm op}(D))\in\mathbb{Z}.$$

Moreover, if $l=2$,

$$\mathrm{DT}^{\rm num}_d(\mathsf{Q}(Y(D)))=|\mathrm{GV}_d(E_{Y(D)})|.$$

This correspondence is a proposed identification of genus-zero log, local, open, and, when $l=2$, quiver Donaldson--Thomas theories; the source states that it was proved in the cited work of Bousseau for the cases under consideration.