Non-factorisation of Arf-Kervaire classes through ${\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$

As an application of the upper triangular technology method of (V.P. Snaith: {\em Stable homotopy -- around the Arf-Kervaire invariant}; Birkh\"{a}user Progress on Math. Series vol. 273 (April 2009)) it is shown that there do not exist stable homotopy classes of ${\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$ in dimension $2^{s+1}-2$ with $s \geq 2$ whose composition with the Hopf map to ${\mathbb RP}^{\infty}$ followed by the Kahn-Priddy map gives an element in the stable homotopy of spheres of Arf-Kervaire invariant one.