The descendent DT4 formula for genus-one Gopakumar–Vafa type invariants

Let $X$ be a smooth projective Calabi–Yau $4$-fold, let $\beta\in H_2(X,\mathbb{Z})$, and let $M_1(X,\beta)$ be the moduli space of one-dimensional stable sheaves $F$ on $X$ with $[F]=\beta$ and $\chi(F)=1$. For $\alpha\in H^2(X,\mathbb{Z})$, use a normalized universal sheaf $\mathbb{F}_{\mathrm{norm}}$ and define $\tau_1(\alpha)=(\pi_M)_*(\pi_X^*\alpha\cup\operatorname{ch}_4(\mathbb{F}_{\mathrm{norm}}))$ and $\langle\tau_1(\alpha)\rangle_\beta=\int_{[M_1(X,\beta)]^{\mathrm{vir}}}\tau_1(\alpha)$. The descendent DT4 conjecture. For a certain choice of orientation on $M_1(X,\beta)$, there is an equality of functions of $\alpha\in H^2(X)$

$$\langle\tau_1(\alpha)\rangle_\beta=\frac{n_{0,\beta}(\alpha^2)}{2(\alpha\cdot\beta)}-\sum_{\beta_1+\beta_2=\beta}\frac{(\alpha\cdot\beta_1)(\alpha\cdot\beta_2)}{4(\alpha\cdot\beta)}m_{\beta_1,\beta_2}-\sum_{k\geqslant1,\ k\mid\beta}\frac{(\alpha\cdot\beta)}{k}n_{1,\beta/k},$$

where $n_{0,\beta}(-)$ and $n_{1,\beta}$ are the genus-zero and genus-one Gopakumar–Vafa type invariants, and $m_{\beta_1,\beta_2}$ are meeting invariants inductively determined by genus-zero invariants. This is the paper's main conjectural sheaf-theoretic interpretation of genus-one invariants; no resolution is reported in the excerpt.