Conjecture on the suspension-kernel classes generated by $[\eta_i]_6$

For each relevant $i$, let $Q_0S^{-2^i+6}$ denote the basepoint component of the infinite loop space of the sphere spectrum shifted by $-2^i+6$, let $[\eta_i]_6\in H_6Q_0S^{-2^i+6}$ be the spherical class corresponding to $\eta_i$, and let

$$\sigma_*:H_*Q_0S^{-2^i+6}\to H_{*+1}Q_0S^{-2^i+7}$$

be the homology suspension.

Suspension-kernel conjecture. The class $[\eta_i]_6$ dies under $\sigma_*$. Consequently, the subalgebra of $H_*Q_0S^{-2^i+6}$ generated by the classes $Q^I[\eta_i]_6$ belongs to $\ker\sigma_*$.

For $i=3$, the paper establishes the corresponding vanishing and uses it to exhibit classes in $H_*Q_0S^{-2}$ that do not arise by pullback from $H_*Q_0S^{-1}$. The general assertion is presented as a conjecture motivated by this case; no resolution is supplied in the source.