The JMGS conjecture for Calabi–Yau threefolds

Let $X$ be a Calabi–Yau threefold. Choose a basis

$$\{\Phi_{\alpha}\}_{\alpha=1}^N=\bigsqcup_{i=0}^3\{\Phi_{ij}\}_{j=1}^{n_i}$$

whose Chern characters form a basis of the corresponding cohomology groups, with $\Phi_{01}=\mathcal{O}$. Let $\{\Phi^{ij}\}$ be the basis dual to $\{\Phi_{ij}\}$ under the K-theoretic Poincaré pairing $(\Phi_a,\Phi_b)^K:=\chi(X,\Phi_a\Phi_b)$. For nonzero curve classes $\beta\in H_2(X,\mathbb{Z})$, write $\langle\frac{\Phi^\alpha}{1-qL}\rangle_{0,1,\beta}$ for the genus-zero, one-point quantum K-theoretic invariant, let $Q^\beta$ denote the corresponding Novikov monomial, and let ${\mathrm{GV}}_{\vec d}$ denote the Gopakumar–Vafa invariant of the curve class $\vec d$. Define

$$a(r,q^r)=\frac{r-1}{1-q^r}+\frac{1}{(1-q^r)^2},\qquad b(r,q^r)=\frac{r^2-1}{1-q^r}+\frac{3}{(1-q^r)^2}-\frac{2}{(1-q^r)^3}.$$

The JMGS conjecture. The small $J$-function satisfies

$$\frac{1}{1-q}J(q,Q)=1+\sum_{\vec d\in H_2(X,\mathbb Z)}\sum_{r=1}^{\infty}\left[\sum_{j=1}^{n_1}\Phi^{1j}\left(\int_{\vec d}\operatorname{ch}(\Phi_{1j})\right)a(r,q^r){\mathrm{GV}}_{\vec d}Q^{r\vec d}+\Phi^{01}b(r,q^r){\mathrm{GV}}_{\vec d}Q^{r\vec d}\right],$$

where

$$\frac{1}{1-q}J(q,Q):=\frac{1}{1-q}\left[(1-q)+\sum_\alpha\sum_{\beta\ne0}\Phi_\alpha\left\langle\frac{\Phi^\alpha}{1-qL}\right\rangle_{0,1,\beta}Q^\beta\right].$$

This generalizes the proposed relation of Jockers, Mayr, Garoufalidis and Scheidegger between Gopakumar–Vafa invariants and quantum K-invariants from the quintic threefold to general Calabi–Yau threefolds. The relation is expressed through the small $J$-function and predicts a precise multiple-cover formula, but its status is not established in the supplied source context.