As a French speaker I had never heard of this, but I looked it up and it’s indeed the case specifically for circumflex accents (ê, ô, â, î) and not the others.

A neat resource (in French naturally) that I found on this:

https://vitrinelinguistique.oqlf.gouv.qc.ca/23698/lorthographe/accents-trema-et-cedille/accent-circonflexe/alternance-entre-laccent-circonflexe-et-le-s-dans-les-mots-de-meme-famille