Extendability of the $B_2$-arrangement

Let $(\mathcal{A},m)$ be a free multiarrangement, and let $\mathcal{E}$ be an extension of $(\mathcal{A},m)$. It is well known that if $\mathcal{A}$ is the Coxeter arrangement of type $A_2$, then a free extension of $(\mathcal{A},m)$ always exists. In this work, we demonstrate that if $\mathcal{A}$ is the Coxeter arrangement of type $B_2$, there exist infinitely many multiplicities for which no free extension of $(\mathcal{A},m)$ exists. This result has immediate consequences for the existence of free extensions in higher rank.