The delta-nine conjecture for ternary sextics

Let $P$ be a nonnegative ternary sextic, let $\Delta_{3,6}$ denote the cone of such forms that are not sums of squares, and let $\delta(P)$ denote its delta invariant. A form is stubborn when all positive odd powers remain nonnegative and are not sums of squares. Delta-nine conjecture. If

$$P\in\Delta_{3,6},\qquad \delta(P)=9,$$

then $P$ is stubborn and lies in the relative interior of a $2$-dimensional face of $P_{3,6}$. The conjecture is motivated by Stengle's form and the known behavior of extremal ternary sextics; no resolution is given here.