Local differential precision operator conjecture for Gaussian Markov random fields on metric graphs

Let $\Gamma$ be a compact metric graph with edge set $\mathcal{E}$. A Gaussian Markov random field of order $p$ is assumed to satisfy some mild regularity conditions. On each edge $e eqinmathcal{E}$, consider a differential operator with coefficients $kappa_{i,e}$, and let $M$ be a finite-rank operator describing the action at the vertices. Local precision operator conjecture. Any Gaussian Markov random field $u$ of order $p$ can be obtained as a Gaussian random field whose precision operator acts on each edge as

$$Q = \sum_{i=0}^{2p} \kappa_{i,e} \frac{d^{i}}{dx_e^{i}},$$

where the coefficient functions $\kappa_{i,e}$ are sufficiently nice, and whose action at the vertices is given by $M$, with $Q$ self-adjoint and local and having compact resolvent. This proposes a general local-operator representation for Gaussian Markov random fields on compact metric graphs; it remains open under the stated mild regularity assumptions.