Realization of the cofibrations as cofibrations of spaces

The preceding constructions involve spectra $\mathbf L(i)\wedge \mathbf T(2^{n-i}-1)$, together with the spectra appearing in the complex of Theorem; the final cofibration has source and target $\mathbf L(n)$ and $\mathbf L'(n)$. A cofibration of spaces means a cofibration in an appropriate category of spaces whose suspension spectrum gives the corresponding cofibration of spectra. Realization conjecture. The preceding cofibrations can be realized as cofibrations of spaces, except for the last one, which can be realized after one suspension and is a retract of the suspension of a cofibration of spaces. The question concerns whether the spectral realization can be obtained geometrically at the level of spaces. The authors report that preliminary computations are positive but that the technical question remains undecided.