Mimura's spherical resolution conjecture for $E_7/F_4$

Localize all spaces at $3$. Let $B_2(27,35)$ denote the space occurring in the proposed spherical resolution, and let $\alpha_2$ be the element generating $\pi_7(S^0)_{(3)}\cong\mathbb{Z}/3$. Mimura's conjecture. The space $E_7/F_4$ is spherically resolved by spheres of dimensions $19$, $27$, and $35$ with attaching maps $\alpha_2$: there are fibrations

$$S^{19}\to E_7/F_4\to B_2(27,35)$$

and

$$S^{27}\to B_2(27,35)\to S^{35},$$

where the attaching maps from $19$ to $27$ and from $27$ to $35$ are both equal to $\alpha_2$. This proposed resolution would give a concrete description of $E_7/F_4$ useful for analyzing its $v_1$-periodic homotopy groups and the associated unstable Novikov spectral sequence; its resolution status is not determined by the supplied text.