Consider a homological localization $L$ of $p$-local spaces or (I think equivalently) a localization of $p$-local spectra. If a space $X$ is $L$-acyclic, then so is $K(\pi_n(X),n)$ for each $n$. To what extent does the converse hold? That is, if $X$ is a space and $K(\pi_n(X),n)$ is $L$-acyclic for each $n$, then is $X$ $L$-acyclic?
If the converse always holds, then it's not known, because this would provide an affirmative answer to the telescope conjecture. So I suppose I should ask:
Question: What is an example of a pair of homological localizations $L,L'$ of $p$-local spaces which agree on Eilenberg-MacLane spaces but not on all spaces?
If $L,L'$ are not $H\mathbb Z_{(p)}$-localization, then they kill some $K(\mathbb Z/p,n+1)$ and thus (if $B\mathbb Q$ is acyclic) all $K(A,m)$'s for $m \geq n+2$. In this case, I believe that $L,L'$ agree on $n+1$-truncated spaces if and only if they agree on Eilenberg-MacLane spaces. On the other hand, if $X$ is $n+1$-connected, then the value of $LX$ or $L'X$ depends only on the $m$-connective cover $\tau_{\geq m} X$ for any $m \geq n+2$. So any difference between $L,L'$ is kind of hard to think about in terms of homotopy groups.
