Curtis–Madsen conjecture on the mod 2 Hurewicz homomorphism

Let $QS^0$ denote the basepoint component of the stable homotopy space of spheres, and let $f\in\pi_{>0}QS^0$ be a stable map of positive degree. The Curtis–Madsen conjecture. The map $f$ is not in the kernel of the mod $2$ Hurewicz homomorphism if and only if $f$ has Hopf or Kervaire invariant one. This conjecture aims to characterize the positive-degree stable homotopy classes detected by the mod $2$ Hurewicz homomorphism; the source recalls it as a conjectural description and gives no resolution status.