Let $G$ and $H$ be torsion abelian groups.
Are the following are equivalent:

1.  $\mathrm{Hom}(G, H) = 0$

2.  $\mathrm{map}_*( K(G, m), K(H,n)) \sim *$ for all $n, m\geq 1$

?

Clearly (2) implies (1).