Local Pandharipande–Thomas invariants equal local Gopakumar–Vafa invariants

Let $X$ be as in the definition of local Gopakumar–Vafa invariants, let $\beta$ be an effective one-cycle class, and let $\gamma\in\operatorname{Chow}_{\beta}(X)$. For $g\in\mathbb Z$, let $n_{g,\gamma}^{P,\mathrm{loc}}$ and $n_{g,\gamma}^{\mathrm{loc}}$ denote the local stable-pair and local Gopakumar–Vafa invariants. Local PT/GV correspondence. One should have

$$n_{g,\gamma}^{P,\mathrm{loc}}=n_{g,\gamma}^{\mathrm{loc}},\qquad \gamma\in\operatorname{Chow}_{\beta}(X),\quad g\in\mathbb Z.$$

This is the local form of the PT/GV correspondence and would identify the two constructible-function-valued theories; the source gives no resolution.