@HarryGindi @DenisNardin Thanks for the paper and for remembering to ping my username. Harry, are you referring to lemma 2.6.4 that says that for an F left exact (F-equivalences, F-local) is a modality? That would go in the wrong direction, though :/ Feel free to say that I haven't checked the paper properly and that, indeed, there's a result going in the correct direction (something holds for F, then F is left exact).

@user40276 I don't know if it's in the public work yet, but Biedermann gave a condition for a modality to be lex in the talk he gave

You could try emailing and asking

perfect timing lol

Is there a "Morava E theory of infinite height" meaning an $E_infty$ ring which is maybe some sort of completion of $BP$ whose bousfield class is the wedge of all morava $K$-theories (not including $K(\infty)$).