Hertling's canonical-structure Torelli conjecture for marked singularities

Fix a singularity $f_0$ and let $M_\mu^{mar}(f_0)$ be the moduli space of marked singularities in its $\mu$-homotopy class. Let $D_{BL}(f_0)$ be the classifying space for Brieskorn lattices, and let

$$BL:M_\mu^{mar}(f_0)\to D_{BL}(f_0)$$

be the holomorphic period map. Equip $M_\mu^{mar}(f_0)$ with its canonical complex structure.

Canonical Brieskorn-lattice Torelli conjecture. The map $BL$ is injective.

The preceding result establishes immersion for the reduced complex structure, while this conjecture asks for global injectivity using the canonical complex structure. The supplied passage gives no resolution status.