Polynomial ideal conjecture for pairwise coprime linear forms

Let $f_1,\ldots,f_m\in\mathbb C[x,y]$ be linear forms such that $\gcd(f_i,f_j)=1$ for all $i\neq j$. For each pair $i<j$, let $\langle f_i+z,f_j+z\rangle$ denote the ideal of $\mathbb C[x,y,z]$ generated by $f_i+z$ and $f_j+z$. Polynomial ideal conjecture. One has

$$\prod_{i=1}^m f_i\notin\bigcap_{1\leq i<j\leq m}\langle f_i+z,f_j+z\rangle.$$

The paper states this as an equivalent commutative-algebraic formulation of the complex arrangement conjecture. It remains open in the source, although the geometric conjecture is verified there for arrangements with at most $12$ lines.