Minimality of the cones produced by the Newton–Puiseux algorithm

Let $\mathbb{K}$ be the coefficient field, let $\bold{x}$ denote the tuple of auxiliary variables, and let $p\in\mathbb{K}[\bold{x}][y]$ be a primitive polynomial. Let $k\in\mathbb{N}$ be the integer input to Algorithm NPA, and let $C$ denote any cone output by the algorithm. Minimal-cone conjecture. If $k$ is large enough, then the cones $C$ output by Algorithm NPA are minimal. The conjecture asserts that sufficiently large input precision makes every cone produced by the algorithm minimal; the supplied source gives no evidence that it has been resolved.