The 37-hyperplane conjecture for irreducible simplicial arrangements

Let $\mathcal{A}$ be an irreducible simplicial arrangement of rank three. Write

$$\mathcal{R}(1)=\{\mathcal{A}(2n,1)\mid n\geq 3\},\qquad \mathcal{R}(2)=\{\mathcal{A}(4m+1,1)\mid m\geq 2\}.$$

37-hyperplane conjecture. If $\lvert\mathcal{A}\rvert>37$, then

$$\mathcal{A}\in\mathcal{R}(1)\cup\mathcal{R}(2).$$

The claim would show that every irreducible simplicial arrangement of rank three with more than $37$ hyperplanes belongs to one of the two known infinite series, leaving only the observed sporadic arrangements below this bound. The completeness of the rank-three list remains conjectural.