Grothendieck–Gillet–Soulé height-positivity conjecture for the Gross–Schoen cycle

Let $k$ be a number field or a function field of positive characteristic, and let $\Delta_\xi$ be the canonical Gross–Schoen cycle associated with the relevant curve or fibration. Grothendieck–Gillet–Soulé conjecture. The height pairing satisfies

$$\langle\Delta_\xi,\Delta_\xi\rangle\geq0.$$

Moreover, equality holds precisely when $\Delta_\xi$ is rationally equivalent to $0$. This height inequality is the missing ingredient needed in the paper's arguments to extend the effective Bogomolov results to positive characteristic; the source gives no resolution status.