Delooping conjecture for the reduced bu Whitehead complex

Work at a prime $p$, and consider the connecting maps in the Taylor tower evaluated at ${\mathbb C}^0$:

$$\Omega^{k-1}\Omega^\infty{\overline{M}}(k)\longrightarrow B\Omega^k\Omega^\infty{\overline{M}}(k+1)\simeq\Omega^{k-1}\Omega^\infty{\overline{M}}(k+1).$$

A contracting homotopy for the complex of spectra ${\overline{M}}(k)$ would follow from suitable deloopings of these maps. Delooping conjecture. $(k-1)$-fold deloopings of the connecting maps give a contracting homotopy for the reduced $bu$ Whitehead complex.

The conjecture links the reduced $bu$ Whitehead problem to the Weiss Taylor tower for the functor $V\mapsto BU(V)$. The paper explains why the connecting maps deloop $k-1$ times, but does not prove that the resulting deloopings give a contracting homotopy.