The prime-power vanishing conjecture for algebraic monodromy of arrangements

Let ${\pazocal{A}}$ be an arrangement of rank at least $3$. For each prime $p$ and integer $s\ge 1$, let ${e}_{p^s}({\pazocal{A}})$ denote the corresponding elementary divisor multiplicity, and let $\beta_p({\pazocal{A}})$ denote the modular resonance invariant. Prime-power vanishing conjecture. One has

$${e}_{p^s}({\pazocal{A}})=0$$

for all primes $p$ and integers $s\ge 1$, with the two possible exceptions

$${e}_2({\pazocal{A}})={e}_4({\pazocal{A}})=\beta_2({\pazocal{A}})\qquad\text{and}\qquad {e}_3({\pazocal{A}})=\beta_3({\pazocal{A}}).$$

The conjecture predicts that only the primes $2$ and $3$ can contribute nontrivially to the algebraic monodromy of the Milnor fibration. The paper reports this pattern in many examples, including rank-$3$ simplicial arrangements, and notes that no arrangement of rank at least $3$ is known with nonzero $\beta_p({\pazocal{A}})$ for $p>3$; the general assertion remains open.