The vanishing proportion conjecture for magic $n_3$-configurations

An $n_3$-configuration is a $3$-uniform configuration with $n$ points and $n$ lines, in which every point lies on three lines and every line contains three points. Call such a configuration magic over an abelian group if its incidence matrix admits a labeling by elements of the group satisfying the magic condition on every line. Vanishing proportion conjecture.

$$\lim_{n\to\infty} \frac{\text{\# of $n_3$-configs magic over some group}}{\text{\# of $n_3$-configs}} \to 0.$$

The conjecture is suggested by computations for the known complete lists of $n_3$-configurations with $7\leq n\leq 14$ and by the increasing prevalence of forbidden substructures, including collections of parallel lines, as $n$ grows.