The generalized Smale conjecture for spherical three-manifolds

Let $(M,g)$ be a closed orientable Riemannian three-manifold, with $g$ a metric of constant sectional curvature $\pm1$. Let $\operatorname{Isom}(M)$ be the isometry group of $M$ and $\operatorname{Diff}(M)$ its diffeomorphism group. Generalized Smale conjecture. The inclusion

$$\operatorname{Isom}(M)\hookrightarrow\operatorname{Diff}(M)$$

is a homotopy equivalence. The paper proves this statement for $\mathbb{RP}^3$ and all lens spaces, while the general formulation remains open.