Cohen–Lenstra-inspired conjecture on p-rational cyclic extensions

Fix a prime $p\geq 5$ and an integer $m>1$ not divisible by $p$. Let $K_0$ be a $p$-rational quadratic imaginary field, and let $\mathcal F_m(K_0)$ be the family of cyclic extensions of degree $m$ over $K_0$, ordered by increasing absolute discriminant. Positive-proportion p-rationality conjecture. The proportion of fields $K$ in $\mathcal F_m(K_0)$ that are $p$-rational is positive. This conjecture is motivated by the Cohen–Lenstra heuristics and is presented as heuristic evidence for the existence of suitable $p$-rational fields; no resolution is given.