A degeneration of the $q$-Garnier system of fourth order arises from confluences in quivers

The $q$-Garnier system was first proposed by Sakai and its other directions of discrete time evolutions were given by Nagao and Yamada. Recently, it was shown that all of those directions of discrete time evolutions are derived from a birational representation of an extended affine Weyl group which arises from the cluster algebraic construction established by Masuda, Okubo and Tsuda. In this article, we investigate a degeneration structure of the $q$-Garnier system of fourth order by using confluences in quivers.