The reciprocity-image lower-bound conjecture for CM fields

Let $K$ be a CM field of degree $2g$, and let $S$ be a primitive CM type for $K$. Let $L$ be the normal closure of $K$, and let $r_K:L^{\times}\to K^{\times}$ be the reciprocity morphism corresponding to $S$. Let $\widetilde{r_K}:\operatorname{Cl}(L)\to\operatorname{Cl}(K)$ be the induced map on class groups. Reciprocity-image conjecture. There exists $\delta(g)>0$ such that

$$|\operatorname{im}(\widetilde{r_K}:\operatorname{Cl}(L)\to\operatorname{Cl}(K))|\gg_{g,\epsilon}\operatorname{Disc}(K)^{\delta(g)-\epsilon}.$$

A positive answer would provide the field-theoretic input needed for the corresponding lower bound for Galois orbits of simple CM abelian varieties with full endomorphism ring. The source presents this as a necessary conjectural reduction and gives no resolution.