Simple homotopy conjecture for crosscut complexes of lattices

Let ${\mathcal L}$ be a lattice and let $C$ be a crosscut of ${\mathcal L}$. Write $\Gamma(C,{\mathcal L})$ for the crosscut complex, $\operatorname{Bd}(\Gamma(C,{\mathcal L}))$ for its barycentric subdivision, and $\Delta(\bar{\mathcal L})$ for the order complex of the proper part of ${\mathcal L}$. Crosscut simple homotopy conjecture. The simplicial complexes $\operatorname{Bd}(\Gamma(C,{\mathcal L}))$ and $\Delta(\bar{\mathcal L})$ have the same simple homotopy type. This extends the weak form of the crosscut theorem from atom crosscuts to arbitrary crosscuts; the source presents it as a conjecture, and no resolution is supplied here.