All groups here are abelian and $p$ is a prime number; I'll say $P$ is a $p$-group if every element of $P$ has finite order which is a power of $p$.

Suppose $\mathrm{Hom}(G,P) = 0$ for every $p$-group $P$. Does it follow that $\mathrm{map}_*(K(G,n), K(P,m))\sim *$ for all $p$-groups $P$ and all $m,n\geq 1$?

Certainly it is true if $G$ is a finitely generated group or an arbitrary torsion group (all elements of finite order); and it is true for any $G$ in the special case $m = n$. But I worry about the possibility of some oddball group with elements of infinite order such that $[ K(G,n), K(P,n+k)] \neq *$ for some $k > 0$ and some $p$-group $P$.