It seems to me like this statement is folklore, since e.g. the paper [Stellar Stratifications on Classifying Spaces][https://arxiv.org/abs/1804.11274v2] tries to show a generalization of it and at least hints that my simpler claim is true (see 1.9). I think I have found a simple proof, but would be happy about corrections/ comments.
First, note that a regular CW complex $X$ is in particular normal. In particular, the set of cells $S$ carries a partial order where $e_1 \leq e_2$ iff, equivalently,
- $e_1 \subseteq e_2$