The DT/PT correspondence for Calabi–Yau 4-folds

Let $X$ be a smooth projective Calabi–Yau 4-fold, let $I_n(X,\beta)$ be the Hilbert scheme of one-dimensional subschemes with Chern character $(0,0,0,\beta,n)$, and let $I_{n,\beta}(X)(\gamma_1,\ldots,\gamma_m)$ and $I_{0,\beta}(X)$ be the associated ideal-sheaf invariants. Let $P_{n,\beta}(X)(\gamma_1,\ldots,\gamma_m)$ and $P_{0,\beta}(X)$ be the corresponding stable-pair invariants. DT/PT correspondence. For $\beta\in H_2(X)$, $\gamma_i\in H^*(X,\mathbb Z)$, and $n\in\mathbb Z$, there exists a choice of orientation such that

$$I_{0,\beta}(X)=P_{0,\beta}(X),\qquad I_{n,\beta}(X)(\gamma_1,\ldots,\gamma_m)=P_{n,\beta}(X)(\gamma_1,\ldots,\gamma_m).$$

This is the proposed Calabi–Yau 4-fold analogue of the established DT/PT correspondence for Calabi–Yau 3-folds. It is verified in several examples, but remains open in general.