Strong concordance-implies-isotopy conjecture for positive scalar curvature metrics

Let $\mathrm{SingMet}$ and $\mathrm{ConcSet}$ be the cubical models of, respectively, the space of positive scalar curvature metrics and the space of positive scalar curvature concordances, and let

$$\mathrm{susp}_\bullet\colon \mathrm{SingMet}\dashrightarrow\mathrm{ConcSet}$$

be the cubical map sending a singular metric to its suspension. Strong concordance-implies-isotopy conjecture. The cubical map

$$\mathrm{susp}_\bullet\colon \mathrm{SingMet}\dashrightarrow\mathrm{ConcSet}$$

is a weak homotopy equivalence. This is a strengthening of the path-component formulation of concordance-implies-isotopy: it asks for equivalence on all homotopy groups, not only injectivity on components.