Comparing constructions of the classifying space for the fibre of the double suspension (Q2161366)

In an uncited paper [\textit{B. Gray}, Trans. Am. Math. Soc. 340, No. 2, 617--640 (1993; Zbl 0820.55006); ibid. 340, No. 2, 595--616 (1993; Zbl 0820.55005)], it was conjectured that the fiber \(W_n\) of the secondary suspension of Mahowald and Cohen is homotopy equivalent to \(\Omega^2T_{(2np-1)}\), where the spaces \(T_k\) were conjectured \(k-1\) connected homotopy abelian \(H\) spaces forming a spectrum equivalent to the Moore space spectrum localized at \(p>3\). It had been previously proved [\textit{B. Gray}, Topology 27, No. 3, 301--310 (1988; Zbl 0668.55005)] that \(W_n\) was homotopy equivalent to the loops on a space \(BW_n\). The spaces \(T_k\) were first constructed by \textit{D. Anick} [Differential algebras in topology. Wellesley, MA: A. K. Peters (1993; Zbl 0770.55001)], and reconstructed in a much simpler away in [\textit{B. Gray} and \textit{S. Theriault}, Geom. Topol. 14, No. 1, 243--275 (2010; Zbl 1185.55011)], but the homotopy equivalence between \(W_n\) and \(\Omega T_{(2np-1)}\) eluded those who attempted to construct an equivalence. In this article, the authors attempt to construct a different classifying space for \(W_n\). However, although the construction is new, the constructed space is also homotopy equivalent to \(BW_n\) and the issue remains unresolved.