Is there a sort accepted general consensus opinion about whether the K(n)-local homotopy groups of the sphere are finitely generated or not?

as a what

by which i mean, if you mean as abelian groups the answer is they are certainly not.

to illustrate that point, note that pi_0 L_K(n) S is Z_p (at least when p>>n, and I think also at all primes)

@SaalHardali it's suspected that this is so, but it's also suspected to be extremely subtle. this is Problem 16.2 in the Hovey-Strickland manuscript, where you'll find some discussion, and afaik there's been no progress on the question

theorem 15.1 referenced there is one of my favorite theorems in K(n)-local homotopy theory

(sorry: suspected that this is so after correcting for peter's objection and speaking of Z_p-modules instead of abelian groups)

which is a reasonable correction, and it's probably fair in retrospect to assume he meant as Z_p modules

@skd I thought $\pi_0 L_{K(n)} S$ is always $Z_p$ regardless of the prime. Is this also open? Or were you just being careful when you said "i think"?

And yeah i meant as modules over the $\pi_0$ which i thought was $Z_p$ always but now im not so sure

@SaalHardali When n=1 and p=2 it's Z_p + \Z/2

Another relevant paper is by Devinatz - sciencedirect.com/science/article/pii/S0001870808001874

i was being careful since i'd clearly forgotten about the height 1 case at p=2