Generalized Smale Conjecture for constant-curvature 3-manifolds

Let $(M^3,g)$ be a closed Riemannian 3-manifold with constant curvature $K\equiv\pm1$. Let $\operatorname{Isom}(M,g)$ be its isometry group and $\operatorname{Diff}(M)$ its diffeomorphism group, with the natural inclusion

$$\operatorname{Isom}(M,g)\longrightarrow\operatorname{Diff}(M).$$

Generalized Smale Conjecture. This inclusion is a homotopy equivalence. The conjecture was verified by Kleiner and Bamler, giving a complete description of the relevant homotopy type in this constant-curvature setting.