A Uniqueness Property of τ -Exceptional Sequences

Recently, Buan and Marsh showed that if two complete τ -exceptional sequences agree in all but at most one term, then they must agree everywhere, provided the algebra is τ -tilting finite. They conjectured that the result holds without that assumption. We prove their conjecture. Along the way, we also show that the dimension vectors of the modules in a τ -exceptional sequence are linearly independent.