  Let $\phi:G \to G$ be a group endomorphism where $G$ is a finitely generated
group of exponential growth, and denote by $R(\phi)$ the number of twisted
$\phi$-conjugacy classes. Fel'shtyn and Hill \cite{fel-hill} conjectured that
if $\phi$ is injective, then $R(\phi)$ is infinite. This conjecture is true for
automorphisms of non-elementary Gromov hyperbolic groups, see \cite{ll} and
\cite {fel:1}. It was showed in \cite {gw:2} that the conjecture does not hold
in general. Nevertheless in this paper, we show that the conjecture holds for
the Baumslag-Solitar groups $B(m,n)$, where either $|m|$ or $|n|$ is greater
than 1 and $|m|\ne |n|$. We also show that in the cases where $|m|=|n|>1$ or
$mn=-1$ the conjecture is true for automorphisms. In addition, we derive few
results about the coincidence Reidemeister number.
