Is the intersection of any two finitely generated subgroups of $\operatorname{Aut}(F_2)$ (resp. $\operatorname{Aut}(F_3)$) again finitely generated? That is, does $\operatorname{Aut}(F_2)$ (resp. $\operatorname{Aut}(F_3)$) have the Howson property? Here $F_n$ is the free group of rank $n$.

For $n \geq 4$, it follows by @YCor’s comment that $\operatorname{Aut}(F_n)$ does not have the Howson property.

This question is maybe very simple (apologies if so!), but I have not been able to track down any references for this. I suppose my question extends quite naturally to the same question for the outer automorphism group $\operatorname{Out}(F_3)$ given the embedding $\operatorname{Aut}(F_n) \hookrightarrow \operatorname{Out}(F_{n+1})$. Note that $\operatorname{Out}(F_2) \cong \operatorname{GL}_2(\mathbb{Z})$, which is virtually free and hence has the Howson property.