Zero-density conjecture for genus numbers in abelian towers

Let $G$ be a finite group of order $n$, and let $b\mathcal{M}'(G,n,X)d$ denote the subset of $b\mathcal{M}(G,n,X)d$ consisting of number fields $K$ that admit a tower

$$\mathbb{Q}=K_1\subset K_2\subset\cdots\subset K_{k+1}=K,$$

in which each extension $K_{j+1}/K_j$ is abelian for all $1\leq j\leq k$. Zero-density conjecture. For any positive integer $t$, as $X\to\infty$, the set of fields $K\in\mathcal{M}'(G,n,X)$ with $g_K=t$ has density zero. This conjecture identifies iterated abelian extensions as families in which every prescribed genus number should occur with density zero, contrasting with the positive-proportion phenomenon observed for the $S_3\times C_q$ and $D_4$ families. It is motivated by the known zero-density result for abelian number fields, while its validity for the broader class of fields admitting an abelian tower remains open.