Conjecture on polynomial convexity and analytic discs for the surfaces $M_t$

Let $M_t$ be the one-parameter family of real surfaces in $\mathbb{C}^2$ considered in the paper, with $t>0$. A surface is locally polynomially convex at the origin if its intersection with some neighborhood of the origin is polynomially convex in that neighborhood, and its local polynomial hull is the corresponding local polynomial hull.

Polynomial-convexity and analytic-disc conjecture. The surface $M_t$ is locally polynomially convex if $t>1$. The local polynomial hull of $M_t$ contains an analytic disc for $t<1$.

This conjecture proposes a dichotomy between the hyperbolic regime $t>1$, where local polynomial convexity is expected, and the elliptic regime $t<1$, where the local polynomial hull is expected to contain an analytic disc. The cited theorem establishes local polynomial convexity for $t>\operatorname{cosec}(\pi/k)$ when the degeneracy order satisfies $k\geq 3$, but the conjectured full range and the analytic-disc assertion are not resolved here.