Integral affine functions without common factors

Question

Let $f_1$, $\ldots$, $f_m \colon \mathbb{Z}^n \to \mathbb{Z}$ affine functions ( an affine function $f$ is of the form $f(x_1, \ldots, x_n) = \sum_{j=1}^n a_j x_j + b $, with $a_j$, $b \in \mathbb{Z}$), such that for every $x \in \mathbb{Z}^n$ we have

$$\operatorname{gcd} (f_1(x), \ldots, f_m(x) ) = 1$$

Then there exist $(\alpha_1, \ldots, \alpha_m) \in \mathbb{Z}^m$ such that

$$\sum_{i=1}^m \alpha_i f_i = 1$$

Notes:

1. The converse is clear.

2. Here is some a particular case of this. Since all of the problems use the same idea of a linear combination, one is lead to believe that this is the underlying reason in general.

3. It works for $A$ a PID rather than $\mathbb{Z}$. For $A = k$ a field, it is an earlier posted question .

4. One could ask if it works or not for other rings, interesting in itself

Any feedback would be appreciated!

$\bf{Added:}$

The problem as stated is in fact Wrong. However, there are some cases where it seems to hold. For now, I will add the extra hypothesis that every $f_i$ has its coefficients relatively prime ( that is, it is not divisible by another integral affine function).