In a recent question, Tim Campion was interested in analyzing the Morava $K$–theory of a space $X$ by dissecting the space into connective and coconnective parts: $$X(m, \infty) \to X \to X[0, m].$$ It is a consequence of work of Ravenel and Wilson that this dissection is "stable": for a fixed height $d$ and for $m \gg d$, the natural maps $X(m, \infty) \to X[m, \infty)$ and $X[0, m] \to X[0, m)$ induce $K$–equivalences, hence the decomposition is independent of the precise value of $m$. We refer to these respectively as the "top" and "bottom" parts of (the $K$–homology of) $X$.

The bottom part is very amenable to study: for instance, Hopkins, Ravenel, and Wilson give a hefty set of tools for its analysis. The top part is quite a bit more curious, but a handful of things about it are known. As part of exploring the $K$–theory of infinite loopspaces, Kuhn dubs the top part as the "virtual $K$–homology of $X$",^{†} and he shows (among other things) that if $X$ is an infinite loopspace, there is a relationship between its virtual homology and the $K$–homology of the free $E_\infty$–ring on (the connective spectrum associated to) $X$.

In a subsequent question, Tim asked about the behavior of $K$–theory on loopspaces. Although it has a negative answer as stated, that answer draws from this same circle of ideas. Bauer addressed the behavior of applying the loopspace functor to the bottom part: provided that $\pi_{d+1} X = 0$ and that $\pi_* X[0, m]$ is finite, there is a natural convergent spectral sequence $$\operatorname{Cotor}_{*, *}^{K_* X[0, m]}(K_*, K_*) \Rightarrow K_* \Omega(X[0, m]).$$ (The negative answer then comes about from falsifying these hypotheses.)

In an effort to pair these two threads of inquiry, let's grant these two conditions on the homotopy of $X$, but let's then try to analyze $K_* \Omega X$ rather than $K_* \Omega(X[0, m])$. A decomposition of $X$ into connective and coconnective parts induces a decomposition of the loopspace: $$\begin{array}{ccccc}\Omega(X(m, \infty)) & \to & \Omega X & \to & \Omega(X[0, m]) \\ || & & || & & || \\ (\Omega X)(m-1, \infty) & \to & \Omega X & \to & (\Omega X)[0, m-1].\end{array}$$ Bauer's result allows us to visit information about the bottom part of $X$ upon the bottom part of $\Omega X$. What about the respective top parts?

Question:Take $X$ to satisfy $X = X(m, \infty)$ for $m \gg d$. Is there a convergent spectral sequence of signature $$\operatorname{Cotor}_{*, *}^{K_* X}(K_*, K_*) \Rightarrow K_* \Omega X?$$ (That is: given information about $K_* X(m, \infty)$, is there an analogous mechanism for visiting it upon $K_* (\Omega X)(m-1, \infty)$?)

I'm happy for the answer to be "no"; I'm not stellar at computing $\operatorname{Cotor}$, and it would be nice to see a carefully worked counterexample. I’d be happy to assume, as Tim does, that $X$ has a delooping (or several deloopings) if that were to change things. If "yes", then I'd also be happy to hear about the ensuing extension problem, but I wouldn't know what to even formulate about it without understanding the behavior of the highlighted question first. (I’m also happy to get an answer to a question I ought to be asking instead, should a reader be able deduce that from context.)

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^{†} - To head off any confusion, I believe that "virtual homology" is not actually a homology theory: the virtualization of the cofiber (resp. coproduct) need not agree with the cofiber (resp. coproduct) of the virtualizations. I haven't carefully checked the effect of either of these on $K$–theory, though—it could be that all these difference somehow end up being $K$–acyclic, but this would be seriously startling and is not at all my expectation.