A theorem of Hopkins and Mahowald states that the Thom spectrum of the map $\Omega^2 S^3 \to B\mathrm{GL}_1(\mathbb{S}_{(p)})$ classifying the element $p$ is exactly $\mathrm{H}\mathbf{F}_p$. Let $T(1)$ denote the free $\mathbf{E}_1$-ring with $\alpha_1 = 0$ (so that at $p=2$, it is the 2-localization of the $\mathbf{E}_2$(!)-ring spectrum $X(2)$). The following question stemmed from attempting to understand whether there is a $p$-local orientation $T(1) \to \mathrm{H}\mathbf{F}_p$ which is a map of Thom spectra, i.e., if there is an orientation which comes from Thom-ifying a map of spaces $\Omega S^{2p-1} \to \Omega^2 S^3$ over $B\mathrm{GL}_1(\mathbb{S}_{(p)})$.

The element $\alpha_1\in \pi_{2p-3}(\mathbb{S})$ desuspends to an element $\alpha_1'$ of the unstable homotopy group $\pi_{2p}(S^3)$. The unstable element $\alpha_1'$ gets us a map $S^{2p-2} \to \Omega^2 S^3$, and hence a map $\Omega S^{2p-1} \to \Omega^2 S^3$. Does this Thom-ify to an orientation $T(1) \to \mathrm{H}\mathbf{F}_p$? This would follow, I think, if we knew that the map $\Omega^2 S^3 \to B\mathrm{GL}_1(\mathbb{S}_{(p)})$ is an isomorphism on $\pi_{2p-2}$, but I don't know if this is true.

A related question is the following. The map $\Omega S^{2p-1} \to \Omega^2 S^3$ is in turn is adjoint to a map $\Sigma \Omega S^{2p-1} \to \Omega S^3$. The James splitting tells us that the source splits as $\bigvee_{n\geq 0} S^{2n(p-1)+1}$, so we get maps $S^{2n(p-1)+1} \to \Omega S^3$, which are exactly elements of $\pi_{2n(p-1)+2}(S^3)$. Are these elements (nonzero multiples of) the desuspensions of the other $\alpha$-family elements $\alpha_n$?