Asymptotic half-density conjecture for the $(2,2,2)$ neuromanifold over finite fields

Let $p$ range over primes, and let $\mathcal{P}_{(2,2,2),r=2}(\mathbb{F}_p)$ be the set of functions represented by the corresponding shallow neural network over $\mathbb{F}_p$. Write $\delta_{(2,2,2),2}(p)$ for its density in the ambient space.

Asymptotic half-density conjecture. The arithmetic expressive capacity approaches $\frac{1}{2}$ as $p\to\infty$:

$$\lim_{p\to\infty}\delta_{(2,2,2),2}(p)=\frac{1}{2}.$$

The preceding computations for $p=3,5,7,11$ give densities tending toward one half, contrasting with the complex case, where the corresponding neuromanifold contains an open Zariski set whose closure fills the ambient space. The parser provides no evidence resolving the finite-field limit, so its status remains open.