Hertling's stabilizer conjecture for right-equivalence monodromy

Let $f$ be an isolated singularity, let $G^{mar}_{\mathcal R}(f)$ be the marked $\mu$-constant monodromy group generated by families remaining in the right-equivalence class of $f$, and let $H_0”(f)$ be its Brieskorn lattice. Write

$$\operatorname{Stab}_{G_{\mathbb Z}(f)}(H_0”(f)):=\operatorname{Aut}(Ml(f),V^{>-\infty},H_0”(f)).$$

Stabilizer conjecture.

$$G^{mar}_{\mathcal R}(f)=\operatorname{Stab}_{G_{\mathbb Z}(f)}(H_0”(f)).$$

The paper notes that the left-hand side is contained in this finite stabilizer and cites this equality as a conjecture; it also records cases in which it holds. The general statement remains open.