$\DeclareMathOperator\Diff{Diff}$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)
Also 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.

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?