Tseng–You's projective-bundle and relative Gromov–Witten determination conjecture

Let $D$ be a smooth proper Deligne–Mumford stack with projective coarse moduli space, and let $L$ be a line bundle on $D$. Put

$$Y:=\mathbb{P}(L\oplus \mathcal{O}).$$

Write $\pi:Y\to D$ for the natural projection, and let $D_0,D_\infty\subset Y$ be the zero and infinity sections of $\pi$. Projective-bundle and relative Gromov–Witten determination conjecture. The Gromov–Witten theories of $Y$, $(Y,D_0)$, and $(Y,D_0\cup D_\infty)$ are determined by the Gromov–Witten theory of $D$ and $c_1(L)\in H^2(D)$.

This conjecture supplies the auxiliary projective-bundle and relative theory needed to extend the root-stack determination conjecture beyond the case treated in the paper. It is presented as the remaining ingredient for the general argument; no resolution is given in the supplied text.