Eccles conjecture on unstable Hurewicz images

Let $X$ be a path connected $CW$-complex with finitely generated homology. For $n>0$, let $f\in\pi_n^sX\simeq\pi_nQX$, and let

$$h(f):\pi_n^sX\simeq\pi_nQX\to H_*QX$$

be the unstable Hurewicz homomorphism. Eccles conjecture. If $h(f)\neq0$, then the stable adjoint of $f$ is either detected by homology or detected by a primary operation in its mapping cone.

The conjecture gives a geometric description of nontrivial unstable Hurewicz images. The paper relates it to the Curtis conjecture: Eccles's conjecture for $X=P$ implies Curtis's conjecture, while Curtis's conjecture implies Eccles's conjecture for $X=S^n$ with $n>0$.