Conjectures on magnitude invariants of real hyperplane arrangements

Let $\mathcal{A}$ be a real hyperplane arrangement of rank $r$, let $n=\#\mathcal{A}$, and write its reduced magnitude as

$$\operatorname{Mag}(\mathcal{A},q)=\frac{P(q)}{Q(q)}.$$

Let $\mathcal{T}(\mathcal{A})$ denote the tope graph, let $\Phi_m(q)$ denote the $m$th cyclotomic polynomial, let $\chi_\ell(\mathcal{A})$ denote the magnitude Euler characteristic in length $\ell$, let $MH_{k,\ell}(\mathcal{A})$ denote magnitude homology, let $L(\mathcal{A})$ denote the intersection lattice, and let $\beta_{k,\ell}$ denote the corresponding magnitude Betti numbers. Magnitude-arrangement conjectures. The following assertions are conjectured:

1. If $r\geq 3$ is odd, then $\mathcal{T}(\mathcal{A})$ is not vertex-transitive if and only if $\Phi_{2n}(q)$ divides $Q(q)$.
2. If $r\geq 4$ is even, then $\mathcal{T}(\mathcal{A})$ is not vertex-transitive if and only if $\Phi_n(q)$ divides $Q(q)$.
3. $(-1)^\ell\chi_\ell(\mathcal{A})\geq 0$ for sufficiently large $\ell$.
4. $MH_{k,\ell}(\mathcal{A})$ is determined by $L(\mathcal{A})$.
5. $MH_{k,\ell}(\mathcal{A})$ is torsion-free.
6. If $r\geq 2$ and $\mathcal{A}$ is indecomposable, meaning that it cannot be expressed as a direct sum $\mathcal{A}_1\oplus\mathcal{A}_2$ of nonzero arrangements, then $\beta_{0,0}=\beta_{r,n}$.

These are proposed from observations in examples and concern the denominator of the magnitude, eventual sign patterns, magnitude homology, and a symmetry of Betti numbers. The paper presents them as open problems, with no resolution supplied here.