Quadratic successive-maxima conjecture for non-genus class numbers

Let $K$ be a quadratic field of selected signature $s=\pm1$, with discriminant $D_K$, conductor $f_K$, and class number $H$ in the restricted sense. Put $D=|D_K|$ and $N=\omega(D_K)=\omega(f_K)$, the number of ramified primes. Define

$$C_{\varepsilon}(D)=\frac{H}{2^{N-1}(\sqrt D)^{\varepsilon}},\qquad 0<\varepsilon<1.$$

Quadratic successive-maxima conjecture. As $D$ increases from its minimum, the successive maxima of $C_{\varepsilon}(D)$ occur only for prime conductors $f_K=\ell$ with $\ell\equiv s\pmod 4$ (equivalently, $N=1$ and $H$ odd), except possibly for finitely many cases. This property is independent of $\varepsilon$.

The claim is the quadratic specialization of the broader successive-maxima conjecture and is motivated by computations cited in the source. No resolution or proof is supplied.