Miller's Cohen-Lenstra-Martinet density conjecture for cyclic prime-degree fields

Let $K$ be a cyclic extension of $\mathbb{Q}$ of odd prime degree $q$, let $p$ be a prime not dividing $q$, and let $h_K$ denote the class number of $K$. Let $\omega$ be the multiplicative order of $p$ modulo $q$. Miller's conjecture.

$$\mathrm{Prob}(p\nmid h_K)=\prod_{k\geq 2}(1-p^{-k\omega})^{\frac{q-1}{\omega}}.$$

This is a Cohen-Lenstra-Martinet heuristic prediction for the density of class numbers not divisible by $p$ in cyclic extensions of odd prime degree. The source presents it as supported by the cited work and numerical investigation, without stating a proof or resolution.