The combinatorial formula conjecture for arrangement monodromy

Let ${\mathcal{A}}$ be an arrangement of rank at least $3$, and let $\beta_2({\mathcal{A}})$ and $\beta_3({\mathcal{A}})$ denote its modular beta invariants. The degree-$1$ algebraic monodromy of the Milnor fibration has characteristic polynomial

$$\Delta_{{\mathcal{A}}}(t)=(t-1)^{\left| {\mathcal{A}} \right|-1} ((t+1)(t^2+1))^{\beta_2({\mathcal{A}})} (t^2+t+1)^{\beta_3({\mathcal{A}})}.$$

Combinatorial monodromy conjecture. The characteristic polynomial is determined by this formula. The preceding results establish the formula in several cases, including arrangements with restricted multiplicities and the known sharp modular bounds; its validity for arbitrary arrangements of rank at least $3$ remains open.