The Gray–Theriault conjecture for the homotopy type of $BW_n$

Let $p$ be a prime, let $n$ be a positive integer, and let $r$ be a positive integer. Set $T_{2n}=S^{2n+1}\{p^r\}$, and let $BW_n$ be the space occurring in the EHP sequence for the Anick spaces. Write $T_{2np-1}(p)$ for the Anick space with parameter $r=1$. Gray–Theriault conjecture. There is a homotopy equivalence

$$BW_n\simeq \Omega T_{2np-1}(p).$$

This was identified in the source as the only remaining unsettled conjecture from the earlier work on Anick spaces. It predicts an explicit loop-space model for $BW_n$, but the supplied text gives no proof or resolution.