Some context.
This question is aiming to fill gaps in a larger proof, so in a way, it is kind of related to this two other questions (this one and that one) that I asked earlier. But since the problem is formulated in a different way, maybe it will be easier to solve it.
The question.
You can choose $\xi=(\xi_1,\xi_2,\xi_3,\xi_4)\in(\mathbb R\setminus\mathbb Q)^4$ however you want, you just need $\xi_5:=\xi_1\xi_4-\xi_2\xi_3\ne 0$.
Can we choose $\xi$ wisely so that there exists a constant $\gamma>0$ and a constant $c>0$ such that
$$\forall a=(a_1,\ldots,a_6)\in\mathbb Z^6\setminus\{0\},\quad \vert a_1\xi_1+a_2\xi_2+a_3\xi_3+a_4\xi_4+a_5\xi_5+a_6\vert\geqslant\frac c{\Vert a\Vert^\gamma}$$
where $\Vert a\Vert^2=a_1^2+\dots+a_6^2$.
Remarks.
The larger $\gamma$ we find, the weaker the result will be, but the easier it will be to prove. Ideally, I would love to find a $\gamma\in[2,4]$.
Anyway, any hint, lead or reference would be much appreciated.