Generally, different reference managers don't play well together in the same doc and I would recommend against trying to have reference from different tools in the same document (nor would this help much: you'd get separate bibliographies etc.).

The one exception is Mendeley and Zotero, which both, I believe, recognize each others citations (though they're not linked to database entries). This is not extensively tested, though, so while it is good enough for migration, I wouldn't, say, want to rely on it for collaboration.

Correct. I'm not aware of any (automated) way of doing this between any two reference managers (in Word documents; obviously you can in LaTeX since the citation format is standardized).

So is there no way to convert, say, Mendeley citations to Zotero citations (or vice versa) so that the resulting document has only the single type?

To add to what adamsmith has said Mendeley and Zotero use a compatible citation format in Word and LibreOffice, so if you go from Mendeley to Zotero or the other way around, then the new reference manager will be able to edit and update citations in your documents. The only issue is that if you need to change the metadata of cited items (e.g. author names, titles, dates) the changes will not be reflected in the document until you re-cite (by deleting and inserting again) those particular items. Zotero treats Mendeley citations similarly to how it would treat items inserted from a collaborator's personal Zotero library -- it allows you to edit those citations and generate a bibliography with them, but metadata edits are not reflected.

As adamsmith said, though, you wouldn't want to collaborate this way between Mendeley and Zotero as everytime you switch reference managers they will "re-adjust" the citations in the document to better fit each program's needs.