The problem:
Given $\Lambda$ be a full-rank lattice in $\mathbb{R}^n$, which has $\lambda_1 < \lambda_2 < \; ... < \lambda_n$ as successive minima. There exist $\textbf{x}_1, \textbf{x}_2, ..., \textbf{x}_n \in \Lambda$ satisfying $\|\textbf{x}_i\| = \lambda_i \; \forall i$.
Decide whether the matrix $$\textbf{X} = [\textbf{x}_1, \textbf{x}_2, ..., \textbf{x}_n]$$ ($\textbf{x}_i$ are its columns) is always a basis of $\Lambda$.
$\text{ }$
Note: I believe the answer is Not always but I've failed to provide a counterexample.