@T.E.D. Yes, they were all under the same King and I found few sources that indicated that Kerala(the region they migrated into) was forming its own culture(birth of a new language - Malayalam) at that time (
kuzhalmannamagraharam.info/articles/kerala-iyer-history.html). I couldn't find any evidences and I couldn't accept this as the link I've quoted in the question says something else altogether...