Global Torelli conjecture for marked singularities

Fix a reference isolated hypersurface singularity $f_0$. Let $M^{mar}_\mu$ be the moduli space of marked singularities, $D_{BL}$ the classifying space of marked Brieskorn lattices, $BL:M^{mar}_\mu\to D_{BL}$ the period map, and $G_{\mathbb Z}$ the Milnor-lattice automorphism group. Let $M_\mu=M^{mar}_\mu/G_{\mathbb Z}$ and let $LBL$ be the induced quotient period map. For a singularity $f$ in the $\mu$-homotopy class and a marking $\rho$, write $\operatorname{Stab}_{G_{\mathbb Z}}([(f,\pm\rho)])$ for the stabilizer and similarly for the period point. Global Torelli conjecture. The following hold: (a) $BL:M^{mar}_\mu\to D_{BL}$ is injective; (b) $LBL:M_\mu\to D_{BL}/G_{\mathbb Z}$ is injective; and (c)

$$\operatorname{Stab}_{G_{\mathbb Z}}([(f,\pm\rho)])=\operatorname{Stab}_{G_{\mathbb Z}}(BL([(f,\pm\rho)])).$$

The conjecture asks whether Brieskorn-lattice period data determine marked or unmarked singularities and whether the period map detects all stabilizers. The source notes that only the inclusion from the left-hand stabilizer into the right-hand one, together with finiteness of both groups, is clear.