Free paths of arrangements of hyperplanes

We study the free path problem, i.e., if we are given two free arrangements of hyperplanes, then we can connect them by free arrangements or not. We prove that if an arrangement $\mathcal{A}$ and $\mathcal{A}\setminus \{H,L\}$ are free, then at least one of two among them is free. When $\mathcal{A}$ is in the three dimensional arrangement, we show a stronger statement.