A question on K(n) local spectra Let $E$ be Morava E-theory at height $h$ and prime $p$. Is every $H_{\infty}$-ring spectrum over $E$ whose homotopy ring is isomorphic to a $W(F_{p^k})[[v_1, \dots, v_{n-1}]][\beta^{\pm 1}]$ a $K(h)$-local spectrum? Here $k$ is a positive integer and $\beta$ has degree 2. 
 A: This isn't exactly an answer but it's longer than a comment. Say $E$ is associated to the height $h$ formal group $\Gamma$ over $k$. Then $$\pi_0L_{K(h-1)}E = Wk((u_{h-1}))^\wedge_p[[u_1, \dotsc, u_{h-2}]].$$
This carries a height $h-1$ formal group, and maps into the Lubin-Tate ring
$$W(k((u_{h-1}))^{perf})[[u_1, \dotsc, u_{h-2}]]$$
of the base change of the universal deformation of $\Gamma$ over $k((u_{h-1}))^{perf}$. If there's a map of fields $k((u_{h-1}))^{perf} \to k$, then you can further map to the height $h-1$ Lubin-Tate ring, $$Wk[[u_1, \dotsc, u_{h-2}]].$$ For example, $k$ could be the perfect closure of $$\mathrm{colim}_n\,\mathbb{F}_q((x_1))((x_2))\dotsm ((x_n)).$$
Then, I think, you can use Goerss-Hopkins obstruction theory to realize this by a map from $E$ to a height $h-1$ $E$-theory, which is in particular $K(h-1)$-local.
It goes without saying that you can't do this over a finite field. If you want to reduce the height of the formal group, you need to make one of the power series generators invertible, and it doesn't seem like there's a lot of space to do that in the rings you mentioned, especially not in an $H_\infty$ way. But I can't think of how to prove anything.
