Extremal-monomial test-configuration conjecture for isolated hypersurface singularities

Let $f(z_0,z_1,z_2,z_3)$ be an isolated hypersurface singularity. The extremal monomials are the monomials selected from the four sets $I$, $II$, $III$, and $IV$ displayed in the source, one from each set, which determine the coordinate weights. A test configuration is a degeneration generated by a $\mathbb C^*$-action whose central fibre has the required enlarged symmetry. Extremal-monomial test-configuration conjecture. It suffices to consider only test configurations that eliminate one of the extremal monomials in $f$. This conjecture proposes a finite restriction on the test configurations relevant to the stability analysis of isolated hypersurface singularities. The source gives no evidence of a proof or disproof.