Holonomicity conjecture for biresidue matrices

Let $B=(b_{ij})_{i,j=0}^{n-1}$ be a biresidue matrix. It is holonomic if, for every subset $J\subset\{0,1,\dots,n-1\}$ of odd cardinality, the linear span of the rows of the submatrix $B_J=(b_{i,j})_{i,j\in J}$ does not contain the vector $(1,1,\dots,1)\in\mathbb{C}^J$. Holonomicity conjecture. Every biresidue matrix listed in the classification of smoothable cycles is holonomic. By the cited equivalence, this would imply holonomicity of the corresponding log symplectic forms and finite-dimensionality of the associated Poisson cohomology everywhere locally. The claim is motivated by numerical experiments and remains unproved in the source.