Four-line generation conjecture for line-arrangement ideals

Let $L=(\mathcal{P},\mathcal{L},\mathcal{I})$ be a line arrangement on $[n]$, and let $I_L$ and $G_L$ be the line-arrangement ideal and the collection of polynomials defined in the paper. If $L$ has at most four lines, then

Four-line generation conjecture.

$$I_L=\langle G_L\rangle.$$

The claim is supported by the line arrangements that the authors were able to compute, but the supplied text does not report a proof or a resolution for all line arrangements with at most four lines.