Bonjour James,
To supplement on Alan's and Chris's excellent answers, here are two answers (from Chris and Alan ;-) ) on the same topic:
https://progress.lawlessfrench.com/questions/view/peut-etre-qu-ils
https://progress.lawlessfrench.com/questions/view/peut-etre-qu-ils
I hope this is helpful.
Bonne journée !
Peut-être que ... -- May be that ...
Don't overthink it, just accept it.
"When followed by a clause, peut-être requires either the relative pronoun que or inversion."
https://www.lawlessfrench.com/expressions/peut-etre/
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