The Čech cohomology invariance conjecture for the o-minimal spectrum

Let $M\preccurlyeq N$, let $X=\operatorname{Spec}(A)$ be the o-minimal spectrum under consideration, let $Z\subseteq X$ be definably closed, and let $W:=\theta^{-1}(Z)$, where $\theta$ is the map from the spectrum over $N$ to $X$ defined in Lemma~. Čech cohomology invariance conjecture. Then $\theta$ induces an isomorphism between the Čech cohomology of $Z$ and that of $W$. The conjecture is known when $M$ expands a field, and the same conclusion holds when $N$ is an o-minimal expansion of $M$ rather than merely an elementary extension; the general case remains open.