Completeness of the rational-point list on the obstruction curve

Let $F(s,a)$ be the obstruction polynomial, let $\widehat{F}(S,A,Z)$ be its homogenization, and let

$$\overline{C}:\quad \widehat{F}(S,A,Z)=0\subset\mathbb{P}^{2}_{\mathbb{Q}}$$

be the projective closure. The computation lists the $8$ rational points

$$\{(1:2:1),\ (0:1:0),\ (0:0:1),\ (0:6:1),\ (1:0:0),\ (-1:2:1),\ (1:0:1),\ (-1:1:0)\}.$$

Completeness of the rational-point list. These are all rational points on $\overline{C}$, so $\overline{C}(\mathbb{Q})$ consists of exactly these $8$ points. If true, this would turn the height-bounded computational evidence into a complete exclusion of affine rational points with $s>0$ and $s\neq 1$ on the obstruction curve. The source presents the assertion in a conjecture environment, and no proof of completeness is supplied.