The convex hypersurface conjecture for infinitesimal strip deformations

For each arc $\alpha\in\mathscr{A}$, let $\underline{\alpha}$ be a minimally intersecting geodesic representative and let $p_{\alpha}$ be a waist; with widths $m_{\alpha}$, define $\boldsymbol f(X)\subset H^1_{\rho}(\Gamma,\mathfrak{g})$ as the image of the complex of arc systems under the corresponding infinitesimal strip deformations. Convex hypersurface conjecture. There exists a choice of the representatives $\underline{\alpha}$ and waists $p_{\alpha}$ such that, when $m_{\alpha}=1$ for every $\alpha\in\mathscr{A}$, $\boldsymbol f(X)$ is a convex hypersurface in $H^1_{\rho}(\Gamma,\mathfrak{g})$. The conjecture would realize the arc-system complex as part of the boundary of the convex hull of a natural discrete subset of the finite-dimensional cohomology space, before projectivization.