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It is possibly relevant to note that, by definition, $|p|$ is a cellular map. If $e$ is a cell of $|E|$ with its natural CW structure, so too is $|p|(e)$ a cell of $|X|$. Conversely, $|p|^{-1}(e)$ is a disjoint union of cells in $|E|$ for every cell $e$ of $|X|$. I think this could be useful.

It is possibly relevant to note that, by definition, $|p|$ is a cellular map. If $e$ is a cell of $|E|$ with its natural CW structure, so too is $|p|(e)$ a cell of $|X|$. Conversely, $|p|^{-1}(e)$ is a disjoint union of cells in $|E|$ for every cell $e$ of $|X|$. I think this could be useful. As a note to myself (inspired by Maxime’s answer), this is only true by invariance of domain for homeomorphisms $p$, since we can have in general a mapping of a nondegenerate $n$-simplex (corresponding to an $n$-cell in the CW structure) to a degenerate one (not corresponding to any particular cell) which breaks the argument.

However, we do, apparently, have that if $|p|$ is a topological covering map (in either the usual or tweaked sense) then $p$ must be a simplicial covering map. I've been able to show all of the above equivalences and implications myself, but this one I am thrown by.

However, we do, apparently, have that if $|p|$ is a topological covering map (in either the usual or tweaked sense) then $p$ must be a simplicial covering map. I've been able to show all of the above equivalences and implications myself, but this one I am thrown by. I know this should be true since it was commented by Tom and also stated in Lurie's "Kerodon", though it was there stated without proof as "proposition: $?$".

I would appreciate any relevant references, proofs or sketches of proofs for why solutions $s$ should exist in the first place. Under the linked post, references were given to May and to Goerss-Jardine: I skim-read the cited chapters, but could not find anything related to this.

It is possibly relevant to note that, by definition, $|p|$ is a cellular map. If $e$ is a cell of $|E|$ with its natural CW structure, so too is $|p|(e)$ a cell of $|X|$. Conversely, $|p|^{-1}(e)$ is a disjoint union of cells in $|E|$ for every cell $e$ of $|X|$. I think this could be useful.

I would appreciate any relevant references, proofs or sketches of proofs for why solutions $s$ should exist in the first place. Under the linked post, references were given to May and to Goerss-Jardine: I skim-read the cited chapters, but could not find anything related to this.