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Let $X$ be a finite spectrum and $K = K(h)$ be the $h$th Morava $K$-theory at the prime $p$. Then $\dim_{K_\ast} K_\ast X$ is increasing in $h$, and eventually constant at $\dim H_\ast(X,\mathbb F_p)$.

Question: Let $X$ be a finite spectrum. Does there exists a finite spectrum $Y$ which is $K(h)$-locally equivalent to $X$ such that $\dim_{K_\ast}K_\ast Y = \dim_{\mathbb F_p} H_\ast(Y;\mathbb F_p)$?

I think the answer is probably no. One way this could be so comes when $\dim_{K_\ast} K_\ast X = 1$ : if there is a nontrivial element of $\operatorname{Pic}(\operatorname{Sp}_{K(h)})$ which is the localization of a finite spectrum, then the answer is no. But I'm not sure if such elements exist.

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    $\begingroup$ There is such an element when $n =1,p = 2$: the exotic element in the Picard group (that generates a $\mathbb{Z}/2$ in the Picard group) is the $K(1)$-localization of the dual of the `question mark' complex. $\endgroup$
    – Drew Heard
    Commented Sep 22, 2023 at 13:15

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