Some time ago I cam across the paper "What we still don't know about loop spaces of spheres" by Ravenel: https://people.math.rochester.edu/faculty/doug/mypapers/loop.pdf which concerns computing Morava K-theory of loop spaces of spheres. In particular, he is giving a couple of conjectures concerning Spanier-Whitehead duals of Snaith summands, which in result should give an idea on computing the aforementioned Morava K-theory.

I wanted to ask if anybody knows what is the current state of art on these problems?

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    $\begingroup$ At least the strongest of Ravenel's conjectures is false, as first observed by Langsetmo. Brantner, Knudsen and I give some references at the end of Section 6.5 of arxiv.org/abs/1908.11321 $\endgroup$ May 7 at 18:54
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    $\begingroup$ The paper of mine that he cites, which tells you a lot about these dual Snaith summands, and certain telescopes among them (Ravenel's $\tilde K_m$'s), was published in Progress in Mathematics, vol. 196 (2001). My paper has its own intriguing conjectures, alluded to by Ravenel. $\endgroup$ May 8 at 1:35

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