Rational-Weierstrass genus-2 Jacobian prime-order conjecture

For a prime $p>2$, let $P_1(p)$ be the probability that a uniformly randomly chosen integer in the Hasse–Weil interval $[(\sqrt p-1)^4,(\sqrt p+1)^4]$ is prime, and let $P_2(p)$ be the probability that a genus-$2$ curve with a rational Weierstrass point, obtained by choosing $f$ uniformly from the monic square-free degree-$5$ polynomials over $\mathbb{F}_p$, has a Jacobian with a prime number of rational points. Let $c_p$ be the constant in the genus-$2$ Jacobian prime-order conjecture. Rational-Weierstrass prime-order conjecture. Then

$$\lim_{p\to\infty}\left(P_2(p)/P_1(p)-\frac{9}{19}c_p\right)=0.$$

The factor $9/19$ reflects the increased probability of rational $2$-torsion, making prime orders substantially less likely; the claim is heuristic and remains unproved.