Eccles's conjecture on spherical classes and detection by homology operations

Let $X$ be a path connected $CW$-complex with finitely generated homology. For $n>0$, let

$$h(f):{_2\pi_n^s}X\simeq{_2\pi_n}QX\longrightarrow H_*QX$$

be the unstable Hurewicz homomorphism, and suppose that $h(f)\neq 0$. The stable adjoint of $f$ can then be detected either by homology or by a primary operation in its mapping cone.

Eccles conjecture. Under these hypotheses, the stable adjoint of $f$ either is detected by homology or is detected by a primary operation in its mapping cone.

The conjecture gives a proposed criterion for nontrivial spherical classes in $H_*QX$. The source relates subsequent results to this conjecture but provides no resolution, so its status is open.