The relative GW/PT correspondence for elliptic-surface geometries

Let $S$ be an elliptic surface, let $C$ be a curve of genus $h$, and let $S_z$ denote the specified relative divisors in $S\times C$. For curve data $(\beta,n)$, insertions $\lambda_1,\ldots,\lambda_N$, and descendent insertions $\prod_{i=1}^{r}{\mathrm{ch}}_{k_i}(\gamma_i)$, let $Z^{(S\times C,S_z)}_{\mathsf{PT},(\beta,n)}$ and $Z^{(S\times C,S_z)}_{\mathsf{GW},(\beta,n)}$ be the corresponding relative Pandharipande–Thomas and Gromov–Witten partition functions. The barred descendent insertion $\overline{\prod_{i=1}^{r}{\mathrm{ch}}_{k_i}(\gamma_i)}$ denotes the result of applying a universal correspondence matrix. The relative GW/PT correspondence. The PT partition function is the expansion of a rational function in $p$, and after the variable change $p=e^z$ one has

$$Z^{(S \times C, S_z)}_{\mathsf{PT}, (\beta,n)}\left( \lambda_1, \ldots, \lambda_N \mid \prod_{i=1}^{r} {\mathrm{ch}}_{k_i}(\gamma_i) \right)=Z^{(S \times C, S_z)}_{\mathsf{GW}, (\beta,n)}\left( \lambda_1, \ldots, \lambda_N \mid \overline{\prod_{i=1}^{r} {\mathrm{ch}}_{k_i}(\gamma_i)} \right).$$

This is the relative version of the GW/PT correspondence, relating curve-counting theories after a universal transformation of descendent insertions. The supplied text gives no resolution status or evidence beyond presenting the relation as conjectural.