Consider a closed $n-$disk and let $\{U_{a}\}_A$ be a finite open cover. Let $\{V_{\alpha}\}_{A'}$ be another finite open cover such that $V_{\alpha_i}\subseteq U_{\alpha_j}$ for all $i$ and some $j.$ (So $\{V_{\alpha}\}_{A'}$ is a refinement of $\{U_{a}\}_A$.)
I am trying to find the following: $\max_{\{U_{\alpha}\}_A} \min_{\{V_{\alpha}\}_{A'}} \max_{x\in\text{Disk}} |\{\alpha:x\in V_{\alpha}\}|$
My guess is that this equals $n-1$ since that seems to be the answer for dimensions 2 and 3 (via drawing) but I don't have a great way of showing it in general. I'm pretty sure the lower bound is $n-1$ for the same reason that 3 and 4 overlapping sets are needed for dimensions 2 and 3, respectively.
(This was relevant to bounding a degree in the graph of the nerve of a cover for my research, but I realized it doesn't actually solve the overarching problem. I am still curious as to people's thoughts, though)