(First, let me apologize that I asked an unanswered and related question in the stable case.)
The category $\operatorname{sSet_+}$ of pointed simplicial sets is symmetric monoidal closed, i.e. there is an internal hom $\operatorname{Hom}(B,C)\in \operatorname{sSet_+}$ for any two pointed simplicial sets $B$ and $C$. The internal hom is adjoint to the smash product, i.e. $$ \operatorname{hom}_{\operatorname{sSet_+}}(A\wedge B, C) \cong \operatorname{hom}_{\operatorname{sSet_+}}(A, \operatorname{Hom}(B,C)) $$ naturally. Let $n$ be a positive integer. To avoid special cases, say $n\geq 5$.
There is a coskeleton functor $\operatorname{cosk}_n(A)$ inducing an isomorphism on all simplicial homotopy groups in degree $<n$.
Let $A, B$ and $C$ be pointed simplicial sets and suppose that every morphism $$ A\to \operatorname{Hom}(B,C) $$ is zero up to homotopy (i.e. null-homotopic) after applying $\operatorname{cosk}_n$. Does it follow that every (adjoint) morphism $$ A\wedge B\to C $$ is zero up to homotopy after applying $\operatorname{cosk}_n$?
Thank you.