DT/PT correspondence conjecture with descendents and Euler-class corrections

Let $X$ be a nonsingular threefold, let $\beta$ be a curve class, let $\alpha_i\in H^{>0}(X)$, and let $k_i,l_j$ be positive integers. Set $\mathrm{e}=c_1(X)c_2(X)-c_3(X)$ and define

$$M(q)=\prod_{i=1}^{\infty}(1-q^i)^{-i},\qquad \mathfrak{F}_1=q\frac{d}{dq}\log M(-q),$$ $$\mathfrak{F}_1^{(i)}=\left(q\frac{d}{dq}\right)^i\mathfrak{F}_1.$$

For disjoint subsets $J_r\subset\{1,\dots,m\}$, put $\overline{J}=\{1,\dots,m\}\setminus\bigcup_rJ_r$, $\lVert J_r\rVert=\sum_{j\in J_r}(l_j-3)$, $f(J)=\prod_{r=1}^sf(J_r)$, and

$$f(J_r)=(-1)^{|J_r|-1}\mathfrak{F}_1^{(|J_r|-1)}\frac{\lVert J_r\rVert!}{\prod_{j\in J_r}(l_j-3)!}.$$

DT/PT correspondence conjecture. For any such data,

$$\begin{aligned} \mathrm{Z}^{\prime,X}_{\rm DT,\beta}\left(\prod_{i=1}^n\mathrm{ch}_{k_i}(\alpha_i)\prod_{j=1}^m\mathrm{ch}_{l_j}(1)\right) &=\mathrm{Z}^{X}_{\rm PT,\beta}\left(\prod_i\mathrm{ch}_{k_i}(\alpha_i)\prod_j\mathrm{ch}_{l_j}(1)\right)\\ &\quad+\sum_{s=1}^m\sum_{J_1,\dots,J_s}f(J)\,\mathrm{Z}^{X}_{\rm PT,\beta}\left(\prod_{i=1}^n\mathrm{ch}_{k_i}(\alpha_i)\prod_{r=1}^s\mathrm{ch}_{\lVert J_r\rVert}(\mathrm{e})\prod_{j\in\overline{J}}\mathrm{ch}_{l_j}(1)\right). \end{aligned}$$

The formula gives a regularized expression for DT descendents in terms of PT invariants and universal degree-zero correction functions.