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@SamHopkins I believe this is the case if $S_n$ is replaced with polynomial representations of $GL_n$ (by Schur Weyl duality). I suspect the answer for symmetric groups (if it exists) should be quite different.
I think bousfield showed that this stronger version of the converse you stated is true for n=1 (i mean combining what bousfield proved for KU with the telescopic conjecture at height 1)
