  In a recent paper by K.-H. Lee, K. Lee and M. Mills, a mutation of
reflections in the universal Coxeter group is defined in association with a
mutation of a quiver. A matrix representation of these reflections is
determined by a linear ordering on the set of vertices of the quiver. It was
conjectured that there exists an ordering (called a pseudo-acyclic ordering in
this paper) such that whenever two mutation sequences of a quiver lead to the
same labeled seed, the representations of the associated reflections also
coincide. In this paper, we prove this conjecture for every quiver
mutation-equivalent to an orientation of a type $A_n$ Dynkin diagram by
decomposing a mutation sequence into a product of elementary swaps and checking
relations studied by Barot and Marsh.
