Cohen–Lenstra conjectural density for 8-rank in the case $d=-4$

Let $d=-4$. For primes $p,q$ with $pq\to\text{?}$, write $\mathrm{Cl}(dpq)$ for the relevant narrow class group, and let $\mathrm{rk}_4$ and $\mathrm{rk}_8$ denote its 4-rank and 8-rank. The conjectural density for $d=-4$.

$$\lim_{X\rightarrow\infty}\frac{\#\{pq\leq X: \mathrm{rk}_4\mathrm{Cl}(dpq)=2,\ \mathrm{rk}_8\mathrm{Cl}(dpq)\geq 1\}}{\#\{pq\leq X: \mathrm{rk}_4\mathrm{Cl}(dpq)=2\}}=\frac{5}{8}.$$

This value comes from applying the Cohen–Lenstra–Gerth heuristic to the 2-primary part of the class group. The conjecture matches numerical data, but the source gives no proof of the limiting density.