The quantum-period conjecture for class TG del Pezzo surfaces

Let $X_1$ and $X_2$ be del Pezzo surfaces of class TG with the same set of qG-rigid cyclic quotient singularities. Let $H_{X_i}^{\mathrm{ts}}$ denote the subspace of twisted sectors of age less than $1$, and let $\varphi\colon H_{X_1}^{\mathrm{ts}}\to H_{X_2}^{\mathrm{ts}}$ be the obvious identification. Let $\widehat{G}_{\mathfrak{X}_1}$ and $\widehat{G}_{\mathfrak{X}_2}$ be the corresponding regularized quantum periods.

Quantum-period conjecture. If

$$\widehat{G}_{\mathfrak{X}_1}=\widehat{G}_{\mathfrak{X}_2}\circ\varphi,$$

then $X_1$ and $X_2$ are qG-deformation equivalent.

This conjecture proposes that, within the class of class TG del Pezzo surfaces having the same qG-rigid cyclic quotient singularities, the regularized quantum period determines the qG-deformation class.