Consider $\omega+1$ with the interval topology, that is $U\subseteq (\omega+1)$ is open if and only if $U\subseteq\omega$ or $(\omega+1)\setminus U$ is finite.

We write $(\omega+1)^\omega$ for the collection of all functions $f:\omega\to(\omega+1)$. Let $\Box (\omega+1)^\omega$ be the set $(\omega+1)^\omega$ endowed with the box topology, where each factor $(\omega+1)$ carries the interval topology.

For each $f\in(\omega+1)^\omega$ we consider the following clopen neighborhood $U_f$ of $f$: $$U_f = \{g\in(\omega+1)^\omega: g(n) \in [n,\omega] \text{ if } f(n) = \omega \text{ and } g(n) = f(n) \text{ otherwise}\}.$$

Let ${\cal U} = \{U_f: f\in(\omega+1)^\omega\}$. Does $\cal U$ have a refinement of pairwise disjoint clopen sets?