Faithful Galois action on components of moduli of minimal surfaces

Let $\phi \in \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be different from the identity and from complex conjugation. Let $S$ be a minimal surface of general type, and let

$$\mathfrak M:= \bigcup_{x,y \geq 1} \mathfrak M_{x,y}$$

be the coarse moduli space of minimal surfaces of general type.

Faithful-action conjecture. There is a minimal surface of general type $S$ such that $S$ and its conjugate $S^{\phi}$ have non-isomorphic fundamental groups. In particular, $S$ and $S^{\phi}$ are not homeomorphic; hence $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the set of connected components of $\mathfrak M$.

This would turn the known phenomenon of Galois-conjugate surfaces with different fundamental groups into a faithful action on moduli components. The supplied text describes this as work in progress, and gives no resolution of the assertion.