In Lemma 2 of [1], Heath-Brown proves the following (I state a simplified version of a more general result):

Let $\Lambda \subset \mathbb{Z}^2$ be a lattice of determinant $d(\Lambda)$. Then $$\# \{ (x_1,x_2) \in \Lambda: \max_i |x_i| \leq B, \gcd(x_1,x_2) = 1\} \leq 16\left (\frac{B^2}{d(\Lambda)} + 1\right).$$

My question is whether this generalises to arbitrary dimensions.

Does an analogous result hold for lattices in $\Lambda \subset \mathbb{Z}^n$? Namely, is $$\# \{ (x_1,\dots, x_n) \in \Lambda: \max_i |x_i| \leq B, \gcd(x_1,\dots,x_n) = 1\} \leq C_n\left (\frac{B^n}{d(\Lambda)} + 1 \right)$$ for some constant $C_n$?

If it helps, I'm primarily interested in the case $n=3$.

Obviously I'm aware of standard lattice point counting techniques, but these usually give an error term of the shape $O(\text{boundary of the region/first successive minima})$, and I don't know how to control this in my case. So I'm just looking for uniform upper bounds where this term doesn't appear.

[1] Heath-Brown - Diophantine approximation with Square-free numbers