Weak geometric version of the Curtis and Eccles conjectures

Work at the prime $2$, with $H_*$ denoting homology with coefficients in $\mathbb Z/2$. Let $Q_0S^0$ be the base-point component of $QS^0$, and for a path connected space $X$ let $QX$ be its associated infinite loop space. Consider the unstable Hurewicz homomorphisms

$$h:\pi_{*>0}Q_0S^0\to H_*Q_0S^0,$$

and

$$h:\pi_*QX\to H_*QX.$$

Weak geometric version of the Curtis and Eccles conjectures. (i) The image of the first homomorphism is finite. (ii) For a path connected space $X$, the image of the second homomorphism is finite.

These are weakened finiteness predictions for unstable Hurewicz images. The first follows from the Curtis conjecture after the solutions of the Kervaire invariant one and Hopf invariant one problems; the second is motivated by the analogous consequence of Eccles's conjecture for spheres, but its general status is not resolved in the supplied text.