Stable-pair genus-one correspondence conjecture for Calabi–Yau fourfolds

Let $X$ be a smooth projective Calabi–Yau $4$-fold, let $P_{0,\beta}$ be the stable-pair invariants, and let $n_{1,\beta}$ be the genus-one Gopakumar–Vafa-type invariants. Define the MacMahon function by

$$M(q)=\prod_{k\geqslant1}(1-q^k)^{-k}.$$

Stable-pair genus-one correspondence conjecture. For a suitable choice of orientation,

$$\sum_{\beta\geqslant0}P_{0,\beta}q^\beta=\prod_{\beta>0}M(q^\beta)^{n_{1,\beta}},$$

where $P_{0,0}:=1$. This predicts that genus-one Gopakumar–Vafa-type invariants control the degree-zero stable-pair partition function; the paper motivates it through ideal Calabi–Yau fourfolds and local calculations, but the general statement remains open.