Intersection Heuristics for anti-cyclotomic extensions

Let $p$ be an odd prime. Let $K$ be an imaginary quadratic field, let $K_{\mathrm{ac}}/K$ be its anti-cyclotomic $\mathbb{Z}_p$-extension, and let $K_n$ denote its subfield of degree $p^n$ over $K$. Let $L(K)$ be the $p$-Hilbert class field of $K$, namely the maximal unramified abelian $p$-extension of $K$. Fix a finite abelian $p$-group $G$, and restrict $K$ to imaginary quadratic fields whose $p$-part of the ideal class group is isomorphic to $G$.

Intersection Heuristics. The probability that

$$L(K)\cap K_{\mathrm{ac}}=K_n$$

is

$$\frac{\#(\text{elements of $G$ of order exactly }p^n)}{\#G}.$$

Equivalently, this probability is

$$\frac{\#(\text{homomorphisms }G\longrightarrow {\mathbb{Q}}/{\mathbb{Z}}\text{ with image of size }p^n)}{\#G}.$$

The heuristic models the $p$-class group as randomly mapping to $\operatorname{Gal}(K_n/K)$; it is based on computational evidence and is not proved in the source.