Let $A$ be a ring spectrum. Suppose that $A$ has a Kunneth theorem -- i.e. the homology theory $A_\ast : Spt \to GrAb$ is a strong monoidal functor [1].

**Question:** Does it follow that $A$ is a module over Morava $K$-theory $K(h)$ for some prime $p$ and some $0 \leq h \leq \infty$?

An affirmative answer would be a variation on the theorem that every ring spectrum which is a _field_ is a module over some $K(h)$. See e.g. [Lurie's notes](https://people.math.harvard.edu/~lurie/252xnotes/Lecture24.pdf).

[1] I have learned [here](https://mathoverflow.net/questions/435685/if-pi-ast-a-is-graded-commutative-then-is-a-ast-a-lax-monoidal-functor) that it's not so straightforward to say exactly what it means to "have a Kunneth theorem" -- right now I'm just assuming that there is _some_ way to make $A_\ast$ into a strong monoidal functor, but it seems in general there need not a canonical way to make $A_\ast$ into a lax monoidal functor, unless $A$ is homotopy commutative. I think for a start, I'd be happy with an answer which assumes that $A$ is homotopy commutative, and assumes that the strong monoidal structure comes from the lax monoidal structure which exists canonically in this case.