Let $X$ be a topological space and $\mathrm{RO}(X)$ its complete boolean algebra of regular opens. Define well inside relation: $$U\prec V\iff\overline{U}\subseteq V.$$ Let $\mathcal C\subseteq\mathrm{RO}(X)$ be a chain such that (a) any two distinct elements of $\mathcal C$ are comparable w.r.t. $\prec$ and (b) $\mathcal C$ is strictly descending, i.e. for every $U$ in the chain there is $V$ in the same chain with $V\subsetneq U$.

Of course, the intersection of $\mathcal C$ does not have to be non-empty (e.g. take $\{(n,+\infty)\mid n\in\omega\}\subseteq\mathrm{RO}(\mathbf{R})$, with $\mathbf{R}$ the set of reals), but it is always non-empty if the space is compact.

Say that a chain $\mathcal C$ *covers* $\mathcal D$ iff for every $U\in\mathcal C$ there is $D\in\mathcal D$ such that $D\subseteq U$. Further, call $\mathcal C$ *c-minimal* if for every $\mathcal D$ covered by $\mathcal C$, $\mathcal C$ is also covered by $\mathcal D$.

My problem is whether for every non-empty c-minimal chain $\mathcal C$ (in an atomless $\mathrm{RO}(X)$) which satisfies (a) and (b), $\bigcap\mathcal C$ is non-empty.

EDIT: Changed the terminology since, as pointed in the comments, the one I used originally was a bit misleading.