Anti-diagonal equivariant Gromov–Witten partition-function correspondence for the resolved conifold

Let $X$ be the resolved conifold, and let $Z_{X,\rm ad}({\bf t};q;\epsilon)$ and $Z_{\mathbb P^1}({\bf s};q;\epsilon)$ denote the partition functions of the equivariant Gromov–Witten theories of $X$ with anti-diagonal action and of $\mathbb P^1$, respectively. Define

$$V_{\rm ad;\epsilon}:=\mathbb{C}((\epsilon))\otimes\mathbb{C}[[s^{1,0}-\sqrt{-1},s^{2,0},s^{1,1},s^{2,1},\dots;q]].$$

For $\alpha=1,2$ and $k\geq0$, set

$$A_{\rm ad}({\bf s})=\sum_{k,\ell\geq0}\frac{(\sqrt{-1})^{k+\ell+1}}{(k+\ell+1)k!l!}s^{1,k}s^{2,\ell}+\sum_{k\geq0}\frac{(\sqrt{-1})^k}{(k+2)k!}s^{2,k},$$ $$t_{\rm ad}^{\alpha,k}({\bf s})=\sum_{\ell\geq0}b^{\alpha,k,\rm ad}_{\beta,\ell}s^{\beta,\ell}-b_{1,1}^{\alpha,k,\rm ad}+\delta^{\alpha,1}\delta^{k,1}.$$

Anti-diagonal equivariant GW correspondence. The identity

$$Z_{\mathbb P^1}({\bf s};q;\epsilon)=\exp\left(\frac{A_{\rm ad}({\bf s})}{\epsilon^2}\right)Z_{X,\rm ad}({\bf t}_{\rm ad}({\bf s});q;\sqrt{-1}\epsilon)$$

holds in $V_{\rm ad;\epsilon}$. This conjecture gives an explicit relationship between the anti-diagonal equivariant Gromov–Witten theory of the resolved conifold and the extended Toda/Gromov–Witten theory associated with $\mathbb P^1$, motivated by Brini's Ablowitz–Ladik conjecture, the Hodge–GUE correspondence, and the Legendre-type relation between the relevant Frobenius manifolds. The supplied source does not state whether this identity has been proved or disproved.