Local polynomial convexity conjecture for the surfaces $M_t$

Let $M_t$ be the real surface under consideration, parametrized by the parameter $t$. A surface is locally polynomially convex at a point if sufficiently small compact neighbourhoods of that point in the surface are polynomially convex. Local polynomial convexity conjecture. The surface $M_t$ is locally polynomially convex when

$$1<t^2\leq\dfrac{15-\sqrt{33}}{8}.$$

This conjecture is posed in the context of determining local polynomial convexity for the family of real surfaces $M_t$ with isolated singularities; its resolution would clarify the transition between local polynomial convexity and the presence of analytic discs near the singularity.