Proofs of two conjectures of Gray involving the double suspension

In proving that the fiber of the double suspension has a classifying space, Gray constructed fibrations

and

He conjectured that $E^{2}\circ\phi$ is homotopic to the $p^{th}$-power map on $\Omega^{2} S^{2np+1}$ when $p$is an odd prime. Harper proved this is true when looped once. We remove the loop when $p\geq 5$. Gray also conjectured that at odd primes $f$ factors through a map

We show that this is true as well when $p\geq 5$.