The Maslov-index criterion for local polynomial convexity near thin CR singularities

Let $\mathcal{S}$ be a smooth real surface in $\mathbb{C}^2$ and let $p\in\mathcal{S}$ be a CR singularity. Assume that $p$ is non-parabolic and that $\mathcal{S}$ is thin at $p$.

Maslov-index criterion. The surface $\mathcal{S}$ is locally polynomially convex at $p$ if and only if

$${\rm Ind}_{\mathcal{M}}(\mathcal{S},p)\leq 0.$$

Theorem on the positive-index case shows that a positive Maslov index prevents local polynomial convexity, while the paper proves only partial converses when the index is nonpositive. Whether nonpositive index always implies local polynomial convexity remains open.