$\DeclareMathOperator\Diff{Diff}$I am looking for the paper computing the homotopy type of the group $\Diff(\mathbb{R}P^3)$ of diffeomorphisms of the $3$-dimensional real projective space $\mathbb{R}P^3$. I know that the homotopy types of $\Diff(S^3)$ and $\Diff(S^2\times S^1)$ are computed by Allen Hatcher, see also explanations by Aramita Amabel.

It should be mentioned that $\pi_0\Diff(L_{p,q})$ for $p\geq2$ is described by Fransis Bonahon. The homotopy types of groups of diffeomorphisms of lens spaces $L_{p,q}$ with $p>2$ are also described in the book "Diffeomorphisms of Elliptic 3-Manifolds" by Sungbok Hong, John Kalliongis, Darryl McCullough, J. Hyam Rubinstein devoted to Smale Conjecture claiming that the inclusion $\operatorname{Isom}(M) \subset \Diff(M)$ is a homotopy equivalence.

However, for the remaining lens space $L_{2,1} = \mathbb{R}P^3$ I can not find a precise information about the homotopy type of $\Diff(L_{2,1})$.

Thus the question is whether the homotopy type of $\Diff(\mathbb{R}P^3)$ is computed? And if yes, could you please provide a reference to the corresponding paper?

Added after the comment below by Andy Putman: If the Smale Conjecture holds for $\mathbb{R}P^3$, then $\Diff(\mathbb{R}P^3) \simeq \mathrm{Isom}(\mathbb{R}P^3) = O(4)/\{I,-I\}$, see description of isometry groups, e.g. Theorem 2.3 in this paper:

• J. Kalliongis, A. Miller, Geometric group actions on lens spaces, Kyungpook Math. J. 42 (2002), no. 2, 313–344, https://kmj.knu.ac.kr/journal/view.html?uid=1266