On a problem of Ganea about strong category (Q1971130)

The author constructs an example of a space \(X\) with \(H_*(X)\) torsion free and such that both \(X\) and \(X\times X\) have the cone-length (string category) 2. This space \(X\) is a 3-localized Berstein-Hilton space \(Y(3)\), i.e. the cone of a 3-localized map \(S^6 \to S^3\). The question about the existence of such \(X\) is related to a problem posed by \textit{T. Ganea} [Lect. Notes Math. 249, 23-30 (1971; Zbl 0237.55009), Problem 8].