Semidirect products with braid groups and type $F_\infty$

Question

Let $F$ be a group which is strongly type $F_\infty$ in the sense that every subgroup is of type $F_\infty$. Here, type $F_\infty$ means that the group admits a classifying space with compact skeleta.

Consider the semidirect product $B_k\ltimes F^k$ where $B_k$ is the braid group on $k$ strands. Think of its elements as braids with strands labelled with elements of $F$. It is of type $F_\infty$ because $B_k$ and $F$ are of type $F_\infty$ and (semi)direct products of type $F_\infty$ groups are of type $F_\infty$.

Let $H,G<F$ be subgroups. Then the subgroups $B_k\ltimes H^k$ and $B_k\ltimes G^k$ of $B_k\ltimes F^k$ are of type $F_\infty$ because $H$ and $G$ are. Also, the intersection $B_k\ltimes H^k\cap B_k\ltimes G^k$ is of type $F_\infty$ because it is $B_k\ltimes(H\cap G)^k$.

Now let $\alpha\in F^k$. I guess that the intersection $$\alpha(B_k\ltimes H^k)\alpha^{-1}\cap(B_k\ltimes G^k)$$ is of type $F_\infty$. Is it true? Is there an easy argument?

Answer 1

In order to close this question, I cite YCor's answer in his comment above: One can replace $B_k$ with the finite index subgroup $P_k$ and in this case $P_k\ltimes F^k=P_k\times F^k$.