The period-map local-homeomorphism conjecture for $q$-quadratic differentials

Let $\log {\mathbf{S}}_\bigtriangleup$ be a topological log surface with its hat homology group $\widehat{\operatorname{\bf H}}_{}(\log {\mathbf{S}}_\bigtriangleup)$, and let $\operatorname{QQuad}_s(\log {\mathbf{S}}_\bigtriangleup)$ be the moduli space of framed $q_s$-quadratic differentials with the prescribed numerical data. The period map

$$\Pi_s\colon\operatorname{QQuad}_s(\log {\mathbf{S}}_\bigtriangleup)\to \operatorname{Hom}_{R}(\widehat{\operatorname{\bf H}}_{}(\log {\mathbf{S}}_\bigtriangleup),\mathbb{C}_s)$$

assigns to a framed differential its period homomorphism. Period-map conjecture. The period map $\Pi_s$ is a local homeomorphism. This is the local-coordinate assertion underlying the period-map description of moduli of $q$-quadratic differentials and their relation to $q$-stability conditions. The supplied text gives no resolution status.