The narrow/wide 2-class group frequency conjecture for cyclic cubic fields

Let $K$ range over cyclic cubic fields whose $2$-class group has two generators. Write $Cl_2^+(K)$ and $Cl_2(K)$ for the narrow and wide $2$-class groups, respectively.

Narrow/wide frequency conjecture. For each relevant $n$, the proportion of fields satisfying

$$Cl_2^+(K)\cong[2^n,2^n]\cong Cl_2(K)$$

is $3/2^{4n-2}$, while the proportion satisfying

$$Cl_2^+(K)\cong[2^{n+1},2^{n+1}],\qquad Cl_2(K)\cong[2^n,2^n]$$

is $3/2^{4n}$.

These predictions sum to Malle's frequency $15/16^n$ and are supported by the tabulated data, but the general distribution remains conjectural.