Determine the survival of an indeterminacy class in the $MO\langle 10\rangle$ Adams spectral sequence

Determine whether the class $h_0^4\vv(20,1,10)$ survives to the $E_4$-page of the Adams spectral sequence for $MO\langle 10\rangle$, in order to resolve the indeterminacy in the computation of the differential $d_4(\vv(20,5,10))$.

Background

The computation of the $d_4$ differential on $\vv(20,5,10)$ uses a Massey product whose indeterminacy includes $h_0^4\vv(20,1,10)$. The argument proceeds under the assumption that this class survives to the $E_4$-page; if it does not survive, the proof would simplify. Thus, the survival question is left unresolved in the presented calculation.