Čech–de Rham isomorphism for non-Hausdorff manifolds

Let $\textbf{M}$ be a non-Hausdorff manifold constructed according to the stated assumptions and Mayer–Vietoris theorem, and let $M_{ij}$ denote its gluing regions. Suppose that each gluing region $M_{ij}$ is regular-open.

Čech–de Rham conjecture. For every $q\in\mathbb{N}$,

$$\check{H}^q(\textbf{M},\mathbb{R})\cong H^q_{dR}(\textbf{M})\cong H^q_S(\textbf{M},\mathbb{R}).$$

This proposes an equivalence between Čech, de Rham, and smooth singular cohomology for the non-Hausdorff manifolds considered in the paper. The surrounding discussion presents it as a direction for further work, and no proof or resolution is supplied here.