Combinatoriality of lattice sheaf cohomology vanishing

Let $\mathcal{A}$ be a hyperplane arrangement in a vector space of dimension $\ell$, let $L_0$ be its intersection lattice with the minimal element removed, and let $\mathscr{D}$ denote the lattice sheaf associated with the module of logarithmic vector fields. Lattice sheaf cohomology conjecture. The vanishing of the groups

$$H^n(L_0,\mathscr{D})$$

for $0<n<\ell-1$ depends only on the poset $L_0$. This is the reformulation of Terao's freeness conjecture given by the authors, connecting freeness to the cohomology of a sheaf on the intersection lattice. The parser supplies no resolution evidence, so the claim is recorded as open.