$\DeclareMathOperator\Diff{Diff}$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. 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" 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?