Asymptotic exponent-difference conjecture for Coxeter $B_2$ multiarrangements

Let $(\mathcal{A},m)$ be a balanced $2$-multiarrangement with defining polynomial

$$Q(\mathcal{A},m)=x^{m_1}y^{m_2}(x-y)^{m_3}(x+y)^{m_4},$$

where $m_i\geq 3$. Write its exponents as $\exp(\mathcal{A},m)=(d_1,d_2)$ and define the difference of the exponents by $\Delta(\mathcal{A},m):=|d_1-d_2|$. Set

$$n_1:=|m_2-m_1|,\qquad n_2:=|m_4-m_3|.$$

Asymptotic exponent-difference conjecture. There exists a natural number $N\in\mathbb{N}$ such that, whenever $\min\{n_1,n_2\}\geq N$, one has $\Delta(\mathcal{A},m)\leq 1$. This conjecture concerns the remaining classification of exponents for the Coxeter arrangement of type $B_2$; only partial classification is currently known, so the asserted threshold remains open.