Explicit bounds for fixed subgroups of endomorphisms of free products

For an automorphism $\phi$ of a free group $F_n$ of rank $n$, Bestvina and Handel showed that the rank $rkFix(\phi )$ 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.