Relative gerbe duality conjecture for Gromov–Witten theory

Let $\mathcal{Y}$ be a gerbe with dual $\widehat{\mathcal{Y}}$ and associated $\mathbb{C}^*$-gerbe $c$. Let $D\subset \mathcal{B}$ be a smooth irreducible divisor, and define its inverse images by

$$\mathcal{D}:=\pi^{-1}(D)\subset \mathcal{Y},\qquad \widehat{\mathcal{D}}:=\widehat{\pi}^{-1}(D)\subset \widehat{\mathcal{Y}}.$$

Write $GW_g(\mathcal{Y},\mathcal{D})$ for the genus $g$ relative Gromov–Witten generating function and $GW_g((\widehat{\mathcal{Y}},\widehat{\mathcal{D}}),c)$ for the corresponding $c$-twisted relative theory. Relative gerbe duality conjecture. As generating functions, the genus $g$ Gromov–Witten theory of $(\mathcal{Y},\mathcal{D})$ is equal to the genus $g$ Gromov–Witten theory of $((\widehat{\mathcal{Y}},\widehat{\mathcal{D}}),c)$:

$$GW_g(\mathcal{Y},\mathcal{D})=GW_g((\widehat{\mathcal{Y}},\widehat{\mathcal{D}}),c).$$

This is the natural relative extension of gerbe duality from absolute to relative Gromov–Witten theory. The supplied text does not state a resolution, so the relative conjecture remains open.