Cao–Kool's MacMahon formula for tautological insertions on Calabi–Yau fourfolds

Let $X$) be a projective Calabi–Yau $4$-fold and let $L$ be a line bundle on $X$. Define

$$I(L;q)=1+\sum_{n>0}I_n(L)q^n=1+\sum_{n>0}\int_{[\operatorname{Hilb}^n(X)]^{\operatorname{vir}}}c_n(L^{[n]})q^n.$$

Here $M(q)=\prod_{i=1}^{\infty}(1-q^i)^{-i}$ is the MacMahon function. Cao–Kool's conjecture. For some choice of orientations,

$$I(L;q)=M(-q)^{\int_Xc_1(L)\cdot c_3(X)}.$$

This predicts a universal generating series for tautological DT$_4$ invariants of Hilbert schemes of points on projective Calabi–Yau fourfolds. The statement is presented as a conjecture in the source; its resolution is not specified there.