The local characterization conjecture for P-Runge pairs of smooth Whitney jets

Let $P(D)$ be a hypoelliptic partial differential operator on $\mathbb{R}^d$ which is orthogonally degenerate on its linear characteristic cone

$$\operatorname{Char}(P)\neq \{0\}.$$

Moreover, let $F_1\subset F_2$ be closed subsets of $\mathbb{R}^d$. The P-Runge pair conjecture. The pair $F_1,F_2$ is a $P$-Runge pair for smooth Whitney jets if and only if, for all $x\in\mathbb{R}^d$ and $\epsilon>0$, the set $F_2$ does not contain a bounded connected component of

$$(\mathbb{R}^d\setminus F_1)\cap\bigl(B(x,\epsilon)+\operatorname{Char}(P)^\perp\bigr).$$

The conjecture proposes a necessary and sufficient local geometric characterization of the $P$-Runge property for smooth Whitney jets in the orthogonally degenerate hypoelliptic case. The surrounding examples establish necessity in particular situations, while the full equivalence remains open.