  A group $G$ has the $R_{\infty}$--property if the number $R(\varphi)$ of
twisted conjugacy classes is infinite for any automorphism $\varphi$ of $G$.
For such a group $G$, the $R_{\infty}$--nilpotency index is the least integer
$c$ such that $G/\gamma_{c+1}(G)$ still has the $R_\infty$--property. In this
paper, we determine the $R_{\infty}$--nilpotency degree of all
Baumslag--Solitar groups.
