  For an automorphism $\phi$ of a free group $F_n$ of rank $n$, Bestvina and
Handel showed that the rank $rk Fix(\phi)$ of the fixed subgroup is not greater
than $n$ (the so-called Scott conjecture). Soon after Bestvina and Handel's
announcement, their result was generalized by many authors in various
directions. In this paper, we are interested in the fixed subgroups of
endomorphisms of free products, focusing on explicit bounds for their ranks.
