The conjecture that the maps $\phi_n$ and $\psi_n$ are homotopic

Let $X$ be the space in the preceding construction, let $n\geq 2$, and let $\phi_n$ and $\psi_n$ be the two maps defined there from the relevant smash-product construction into $\Omega\Sigma(\Sigma^{n-1}X)$. Homotopy conjecture for $\phi_n$ and $\psi_n$. The maps are homotopic:

$$\phi_n\simeq\psi_n.$$

The preceding proposition shows that they induce the same homomorphism in homology, but the source gives no proof that they are homotopic and states no resolution of this conjecture.