Logarithmic GW/PT correspondence for primary insertions

Let $(Y\mid\partial Y)$ be a toric threefold pair, let $\beta$ be a curve class, let $\bm\mu$ be a vector of partitions, and let $\upalpha$ be an insertion. Write $d_\beta$ for the anti-canonical degree of $\beta$, and let $\ell(\mu_j)$ and $|\mu_j|$ denote the length and size of the $j$th partition. GW/PT correspondence. The PT series $Z_{\sf PT}(Y\mid\partial Y;q\mid\upalpha\mid\bm\mu)$ is the Laurent expansion of a rational function. Under $q=-e^{iu}$, one has

$$(-q)^{-d_\beta/2}\cdot Z_{\sf PT}(Y\mid\partial Y;q\mid\upalpha\mid\bm\mu)_\beta = (-iu)^{d_\beta+\sum \ell(\mu_j)-|\mu_j|}\cdot Z_{\sf GW}(Y\mid\partial Y;u\mid\upalpha\mid\bm\mu)_\beta.$$

This is the logarithmic GW/PT correspondence for primary insertions, asserting both rationality of the PT series and equality with the GW series after the specified change of variables and normalization. The source identifies it as a conjecture previously formulated as Conjecture A; the present paper proves the correspondence for equivariant toric threefold pairs.