The saturation conjecture for additive polynomial mappings

Let $K$ be the global function field in the setup, let $F\in\operatorname{End}_{\mathbb{F}_p}(\mathbb{G}_{a/K}^d)$ be an additive polynomial mapping, and let $\kappa$ and $\delta$ denote its saturation and dynamic degrees. Saturation conjecture. (i) If no non-zero proper connected algebraic subgroup of $\mathbb{G}_{a/K}^d$ is stable under $F$, then

$$\kappa=\delta.$$

(ii) For a short exact sequence of $t$-modules

$$0\to H\to G\to G/H\to0,$$

the invariant satisfies

$$\kappa(G)^{\dim G}=\kappa(H)^{\dim H}\kappa(G/H)^{\dim(G/H)}.$$

The conjecture is intended to determine the saturation invariant for all $t$-modules; the source presents it as unproved and likely relevant to the preceding strengthening of the main theorem.