A problem about successive minima of a lattice in $\mathbb{R}^n$

Question

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.

Answer 1

This is a slight modification of the commonly known counterexample for $n=5$.

Let $a=1.00001$ and let $\Lambda$ be the lattice generated by the matrix $$\begin{pmatrix}a&0&0&0&a/2\\0&a^2&0&0&a^2/2\\0&0&a^3&0&a^3/2\\0&0&0&a^4&a^4/2\\0&0&0&0&1/2\end{pmatrix}.$$ Then the successive minima of $\Lambda$ are $1<a<a^2<a^3<a^4$ but the columns of the matrix $$X=\begin{pmatrix}a&0&0&0&0\\0&a^2&0&0&0\\0&0&a^3&0&0\\0&0&0&a^4&0\\0&0&0&0&1\end{pmatrix}$$ do not form a basis of $\Lambda$.