On Freudenthal theorem, Kahn-Priddy Theorem, and Curits conjecture

We verify Curtis conjecture on a class of elements of ${_2\pi_*^s}$ that satisfy a certain factorisation property. To be more precise, suppose $f\in{_2\pi_n^s}$ pulls back to $g\in{_2\pi_n^s}P$ through the Kahn-Priddy map $\lambda:QP\to Q_0S^0$ such that $g$ projects nontrivially to an element $g'\in{_2\pi_n^s}P_{t(n)}$ with $h(g')=0$ where $h:{_2\pi_*}QP_k\to H_*QP_k$ is the unstable Hurewicz map, and $t(n)=\lceil n/2\rceil$. Then, mod out by elements of ${_2\pi_*^s}\simeq{_2\pi_*}QS^0$ satisfying this property, the Curtis conjecture on the image of $h:{_2\pi_*}QS^0\to H_*QS^0$ holds.