Short product representation and expansion conjecture for ideal class groups

Let $G=\operatorname{cl}(\mathcal{O})$ be the ideal class group of an order $\mathcal{O}$ in an imaginary quadratic field, with cardinality $h$. Let $S_k$ be the sequence of ideal classes associated with the first $k$ eligible prime ideals, and let $d$ denote its density. Write $\mathcal{P}(S_k)$ for the corresponding product-probability space, $\pi$ for the product map into $G$, $\pi_*$ for the induced pushforward of probability measures, and $\mathbb U_A$ for the uniform distribution on a finite set or group $A$. Short product representation and expansion conjecture. For every $d_0>1$, there exist constants $c>0$ and $D_0<0$ such that, if $D\leqslant D_0$ and $S_k$ has density $d\geqslant d_0$, then

$$\pi(\mathcal{P}(S_k))=G,$$

that is, $S_k$ represents $G$, and

$$\left\|\pi_*\mathbb U_{\mathcal{P}(S_k)}-\mathbb U_G\right\|<h^{-c}.$$

This conjecture strengthens conditional results on generators and short product representations for ideal class groups, aiming to show both surjectivity of the product representation and quantitative closeness of the induced distribution to uniformity. The source does not provide evidence resolving it, so its status is open.