Let $A$ be a homotopy ring spectrum. Then the homology theory $A_\ast : Spectra \to GrAb$ lifts to a homology theory valued in $GrMod(\pi_\ast A)$. If $A$ is homotopy commutative, then this functor $A_\ast : Spectra \to GrMod(\pi_\ast A)$ is lax monoidal. But it makes sense to ask for a lax monoidal structure on this functor even if $A$ is not homotopy commutative, so long as $\pi_\ast A$ is graded-commutative. And the answer can be yes --e.g. at the prime 2, Morava $K$-theory $K(n)$ is not homotopy commutative, but $K(n)_\ast$ is strong monoidal.

Question: Let $A$ be a homotopy ring spectrum such that $\pi_\ast A$ is graded-commutative. Then is the functor $A_\ast : Spectra \to GrMod_{\pi_\ast A}$ lax monoidal? Even lax symmetric monoidal?