  Recently, Buan and Marsh showed that if two complete $\tau$-exceptional
sequences agree in all but at most one term, then they must agree everywhere,
provided the algebra is $\tau$-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 $\tau$-exceptional
sequence are linearly independent.
