Let $\mathbf{p}$ be a primitive point in the lattice $\mathbb{Z}^J$ and denote the $J-1$ dimensional vector space $V = \mathbf{p}^{\perp} \subseteq \mathbb{R}^J$. Let $\Lambda' = \mathbb{Z}^J \cap V $, which is a lattice of rank $J-1$.

What I am interested in is : does there always exist an integral basis $\{ \omega_1, ..., \omega_{J-1} \}$ of $\Lambda'$ such that for each $j$, $ |\omega_j | \leq |\mathbf{p}|$?

The statement is definitely true when $J=2$. I was curious for $J>2$. I thought it is maybe true, but I haven't been able to show it yet (or disprove it). I would appreciate any hint/solution! (I apologize in advance if this question happens to be not at the mathoverflow level.)