To what extent is homological localization determined by its values on $K(G,n)$'s?

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.

You ask two different questions. Regarding the first, there are easy counterexamples: e.g. $$\widetilde K(1)_*(S^3) \neq 0$$, but $$\widetilde K(1)_*(K(\pi_n(S^3),n)) = 0$$ for all $$n$$.