Nilpotency bounds for groups of self-homotopy equivalences

Let $Y$ be a finite-dimensional complex. Write $\mathcal{E}_{\#}(Y)$ for the group of self-homotopy equivalences of $Y$ inducing the identity on homology, and $\mathcal{E}_{\#\infty}(Y)$ for the subgroup inducing the identity on all homotopy groups. Let $\mathrm{rscl}(Y)$ and $\mathrm{scl}(Y)$ denote the related cone-length invariants, and let $\mathrm{nil}$ denote nilpotency class.

Nilpotency conjecture.

$$\mathrm{nil}\, \mathcal{E}_{\#}(Y)\leq \mathrm{rscl}(Y)-1 \qquad\text{and}\qquad \mathrm{nil}\, \mathcal{E}_{\#\infty}(Y)\leq \mathrm{scl}(Y)-1.$$

The conjecture seeks nilpotency bounds analogous to the solvability bounds established earlier, and is motivated by bounds in work of Arkowitz and Lupton, Félix and Murillo, and Scheerer and Tanré. The authors state that they have not been able to prove it; the second inequality is also attributed to Scheerer and Tanré.