The twisted Morava theory differential conjecture for $W_7+[H_7]$

Let $W_7$ be the degree-seven integral Stiefel–Whitney class and let $[H_7]$ be a degree-seven cohomology class that acts as a twist of a generalized cohomology theory. The expression $W_7+[H_7]$ is then a twisted degree-seven cohomology class.

Twisted Morava theory differential conjecture. The cohomology class $W_7+[H_7]$ corresponds to a differential in twisted Morava $K$-theory and twisted Morava $E$-theory, where $[H_7]$ acts as the twist.

The claim is motivated by the appearance of elliptic cohomology through the condition $W_7=0$ and by the analogy with the differential determined by $W_3+[H_3]$ in twisted K-theory. The source gives no evidence of resolution, so this conjecture remains open.