Ten-round-zero conjecture for singular extremal ternary sextics

Let $P_{3,6}$ be the cone of nonnegative ternary sextic forms, let $\Delta_{3,6}$ be its relevant cone of forms under consideration, and let $E(P_{3,6})$ denote the set of extremal forms in $P_{3,6}$. For a zero $\pi$ of $P$, let $\delta(P,\pi)$ measure the singularity of that zero, and put

$$\delta(P)=\sum_{\pi\in\mathcal Z(P)}\delta(P,\pi).$$

A zero is round when $\delta(P,\pi)=1$. Ten-round-zero conjecture. If

$$P\in E(P_{3,6})\cap\Delta_{3,6},$$

then $\delta(P)=10$, and either $P$ has ten round zeros or $P$ is the limit of psd extremal ternary sextics with ten round zeros. This conjecture refines the paper’s heuristic that coalescing zeros account for higher-order singularities; its general validity is left open.