Bennewitz's conjecture on higher-order HELP inequalities

Let $M_N$ and $M_D$ be the Neumann and Dirichlet Titchmarsh–Weyl matrices, respectively, and let $\operatorname{Res}(M_N,0)$ and $\operatorname{Res}(M_D,0)$ denote their residue matrices at $0$. Assume that $M_N$ is meromorphic, and hence that $M_D$ is meromorphic, and let $n$ be half the order of the differential expression. Bennewitz's conjecture. If

$$\operatorname{rank}(\operatorname{Res}(M_D,0))+\operatorname{rank}(\operatorname{Res}(M_N,0))=n,$$

then there is a valid HELP inequality associated with the differential expression. The conjecture strengthens the known second-order characterization of valid HELP inequalities and, as stated in the paper, is proved there for general even-order HELP inequalities.