Prime affine direction conjecture for additive codes

Let $p$ be prime. A set of $p^2$ points in the affine space $\mathrm{AG}(4,p)$ is said to determine all points of a projective line if its determined directions contain every point of that line.

Prime affine direction conjecture.

$$\begin{aligned} &\text{(i) Every set of }p^2\text{ points of }\mathrm{AG}(4,p)\text{ determines all points of some line of }\mathrm{PG}(3,p).\\ &\text{(ii) Consequently, every extendable additive }(n,2,d)_{p^2/p}\text{-code admits an additive extension.} \end{aligned}$$

The geometric assertion is intended to imply the coding-theoretic assertion. The paper notes that the corresponding extension property is known in some small cases and fails for broader additive-code families, leaving the prime case unresolved.