Definitely not if we’re talking about recorded words. Lax and its cognates are attested in branches of Proto Indo European which aren’t written until the first millennium CE at the earliest. The oldest attestation of a cognate of lax (lax is the native English firm of the word, lox is borrowed from Yiddish which is in turn borrowed from German Lachs) would be from Tocharian, in which laks meant fish. There are so many words which are attested thousands of years earlier with consistent meaning over time across more branches of the Indo European tree like words denoting family relations, food and drink and other basic vocabulary. (Of course that’s not to say there’s no semantic shift in individual branches, like Ancient Greek φρατηρ meaning ‘brother’ in the sense of a fellow member of a community).

The reason why linguists were interested in lax was because of its consistent form over time and across branches of Indo European and the role it played in the question of the Indo-European homeland - the proposed PIE reconstruction *laks- is pretty much identical to Modern English lax and other cognates, although earlier forms of the word such as Old English leax (ea being pronounced like General American English a in cat followed by the first vowel in father) show that the pronunciation isn’t entirely unchanged.

But thanks for your comment, it prompted some research to make my morning more interesting!