Conjecture on uniform distribution of split-place differences in class groups

Let $p$ be an odd rational prime and let $Q$ be either $\mathbb{Q}$ or $\mathbb{F}_q(t)$, with $p\nmid q(q-1)$. For $K/Q$, define

$$\operatorname{Cl}_K= \begin{cases} \operatorname{Cl}(\mathcal{O}_K)_p,& Q=\mathbb{Q},\\ \operatorname{Pic}^0(C_K)_p,& Q=\mathbb{F}_q(t), \end{cases}$$

and write the group law additively. If a finite place $v$ of $Q$ splits completely in a quadratic extension $K$ as $v_1,v_2$, consider the class-group element $v_1-v_2$.

Uniform-distribution conjecture. As such quadratic extensions are ordered by discriminant, $v_1-v_2$ is distributed uniformly in $\operatorname{Cl}_K$. More precisely, in the number-field case restricted to imaginary quadratic fields, or in the function-field case, the isomorphism classes of pairs $(\operatorname{Cl}_K,v_1-v_2)$ are distributed according to $\mu$.

This refines Cohen–Lenstra-type distributional predictions by tracking distinguished elements arising from split places. The supplied text gives no resolution status for this assertion.