Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank sublattice and let $B \in \mathfrak{ut}(\mathbb{Z},n)$ be a upper triangular basis matrix of $\Lambda$. Is $B$ unique up to the right action of $\mathfrak{ut}(\mathbb{Z},n) \cap GL(\mathbb{Z},n)$ (unimodular upper triangular matrices)?