The conjecture on the vanishing of the third James–Hopf obstruction

Let $f:X\to Y$ be a map with $X=\Sigma X'$ and $Y=\Sigma Y'$. Write $M_f$ for the mapping cone of $f$, and let $J_2(M_f,A)$ and $J_2(M_f,X)$ denote the corresponding second James constructions. Suppose that $[\operatorname{id}_Y,f]=0$, so that

$$J_2(M_f,A)\simeq Y\vee Y\wedge X.$$

Let $j_Y$ and $j_{Y\wedge X}$ be the canonical inclusions of the corresponding wedge summands of

$$J_2(M_f,X)\simeq X\vee X\wedge A.$$

The conjecture. Under these hypotheses, the map

$$\gamma_3=[j_Yf,j_{Y\wedge X}]:Y'\wedge X\wedge X\longrightarrow Y\vee Y\wedge X$$

is null-homotopic.

The conjecture is motivated by the fact that it holds in all special cases encountered by the authors, including attaching maps of the top cell in even-dimensional mod $2^r$ Moore spaces, suspended complex projective planes, and suspended quaternionic projective planes. Its general status is not resolved by the supplied source context.