Pachner complex model for Zwiebach's moduli space

Let $\Sigma$ be a surface of genus $h$ with $n$ boundary circles, whose boundary circles are triangulated into $k_1,\ldots,k_n$ intervals. Pachner-complex conjecture. There exists a CW complex $\Xi(\Sigma,\{k_i\})$, called the Pachner complex, such that its vertices correspond to admissible triangulations of $\Sigma$ inducing the prescribed boundary triangulations; its edges correspond to Pachner moves, namely flips or stellar subdivisions and aggregations; it is homotopy equivalent to Zwiebach's moduli space $\widetilde{\mathcal{M}}_{h,n}$; for $h=0$, it is homotopy equivalent to the component $E_2^{\mathrm{fr}}(n-1)$ of the framed little disk operad, with an induced homology isomorphism

$$f_*\colon H_\bullet(\Xi)\longrightarrow H_\bullet(E_2^{\mathrm{fr}}(n-1));$$

compatible with gluing of cobordisms or operadic composition in $E_2^{\mathrm{fr}}$; its faces correspond to admissible polygonal decompositions into polygons with at least three sides, equipped with a linear chart and a configuration chamber of at least zero floating points, with subfaces described locally by secondary-polytope combinatorics; and, for a cylinder with $k_1=k_2=:k$, the BV cycle $\mathsf{c}_\Delta^k$ represents the generator of

$$H_1(\Xi,\mathbb{Z})\stackrel{f_*}{\simeq}H_1(E_2^{\mathrm{fr}}(1),\mathbb{Z})=\mathbb{Z}.$$

This is the proposed ideal combinatorial model of the moduli spaces governing the two-dimensional higher topological quantum field theory. The source does not state a resolution, and the notion of admissibility is explicitly left as a black box.