Suppose $X$ is a locally path connected topological space and $\mathcal{U}$ is an open cover of $X$ (consisting of path connected sets if we want).

One often wants the intersection $A\cap B$ of pairs of elements $A,B\in \mathcal{U}$ to be path connected, or perhaps stronger, that the intersection of finitely many elements of $\mathcal{U}$ be path connected. This, for instance, is the case in some groupoid versions of the van Kampen theorem (like the one in Peter May's *A Concise Course in Algebraic Topology*). Having covers with this property also simplifies life a bit in the study of shape invariants constructed via nerves of covers. If we don't start with a cover this nice perhaps we can at least get to one by refinement.

**Question:** Is it always possible to find an open cover $\mathcal{V}$ of $X$ refining $\mathcal{U}$ such that the intersection of every pair of elements in $\mathcal{V}$ is path connected (or empty)? Can we do even better and find $\mathcal{V}$ such that the intersection of finitely many elements of $\mathcal{V}$ is path connected?

I'm less confident such refinement is possible for general locally path connected $X$. I'd be perfectly content to assume $X$ is paracompact Hausdorff.