Asymptotic scarcity of Legendre cordial labelings for complete graphs

Let $K_n$ be the complete graph on $n$ vertices. For an integer $m$, let $\mathbb{J}(n,m)$ be the set of primes $p\leqslant m$ for which $K_n$ is a Legendre cordial graph modulo $p$, and let $J(n,m)=|\mathbb{J}(n,m)|$. Asymptotic scarcity conjecture. For any integer $m$,

$$\lim_{n\to\infty}J(n,m)=0.$$

Computations indicate that the number of primes up to a fixed bound for which a complete graph admits a Legendre cordial labeling decreases as the graph order grows, while the stated limit remains unproved.