Curtis conjecture on spherical classes in $H_*QS^0$

Let $QS^0=\operatorname{colim}_i\Omega^iS^i$ be the infinite loop space associated to the sphere spectrum. In positive degrees, consider the unstable Hurewicz homomorphism

$$h:{_2\pi_*^s}\simeq{_2\pi_*}QS^0\longrightarrow H_*QS^0.$$

Curtis conjecture. Only the Hopf invariant one and Kervaire invariant one elements survive under $h$.

This conjecture concerns the spherical classes in the homology of the stable infinite loop space $QS^0$ and predicts that the unstable Hurewicz image at the prime $2$ is accounted for precisely by these two families of elements. The source gives no resolution, so its status is open.