The rational-simplex conjecture for nonempty rational quadratic-form graphs

Let $q\in\mathcal{Q}_n$, and let $G_0(\operatorname{\mathbb{Q}}^n,q)$ be the connected component containing $\operatorname{\textbf{0}}$ in the distance graph associated with $q$. A simplex with rational lengths means a set of vertices whose pairwise distances, measured by $q$, are rational. Rational-simplex conjecture. If $G_0(\operatorname{\mathbb{Q}}^n,q)$ is nonempty, then there exists an $(n+1)$-simplex with rational lengths in $(\operatorname{\mathbb{Q}}^n,q)$. The statement concerns the existence of a maximal-size rational simplex in every nonempty graph of this type; no resolution is supplied in the source.