The cohomological DT/PT correspondence with insertions on Calabi–Yau fourfolds

Let $X$ be a smooth projective Calabi–Yau fourfold and let $\beta\in H_2(X,\mathbb Z)$. For a line bundle $L$ on $X$, write $L^{[n]}$ for its tautological bundle on the relevant moduli space, and let $[I_n(X,\beta)]^{\mathrm{vir}}$ and $[P_n(X,\beta)]^{\mathrm{vir}}$ denote the virtual classes of ideal sheaves and stable pairs. Cohomological DT/PT correspondence with insertions. There exist choices of orientations such that

$$\frac{\sum_n \int_{[I_n(X,\beta)]^{\mathrm{vir}}}e(L^{[n]})q^n}{\sum_n \int_{[I_n(X,0)]^{\mathrm{vir}}}e(L^{[n]})q^n}=\sum_n \int_{[P_n(X,\beta)]^{\mathrm{vir}}}e(L^{[n]})q^n.$$

This conjecture is motivated by the toric K-theoretic correspondence and its cohomological limit; the statement concerns smooth projective Calabi–Yau fourfolds beyond the toric setting.