$\DeclareMathOperator\Diff{Diff}$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 [$\Diff(S^3)$](https://doi.org/10.2307/2007035) and [$\Diff(S^2\times S^1)$](https://www.ams.org/journals/proc/1981-083-02/S0002-9939-1981-0624946-2/S0002-9939-1981-0624946-2.pdf) are computed by Allen Hatcher, see also [explanations by Aramita Amabel](https://math.mit.edu/~araminta/SmaleConjecture.pdf).

It should be mentioned that $\pi_0\Diff(L_{p,q})$ for $p\geq2$ is [described](https://doi.org/10.1016/0040-9383(83)90016-2) 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](https://doi.org/10.1007/978-3-642-31564-0)" 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?

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**Added after the comment below by [Andy Putman](https://mathoverflow.net/users/317/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