Let $K$ be a number field and $\mathcal{P}$ be a prime ideal of $K$. Let $k_{\mathcal{P}}$ be the residue field $\mathcal{O}_K/\mathcal{P}$.

Consider the following construction used very often in cryptography. Fix $n \geq 2$ and let $\pi_{\mathcal{P}}: \mathcal{O}_K ^n \rightarrow k_{\mathcal{P}}^n$ be the reduction modulo $\mathcal{P}$ map. I want to consider the lattices 
$$L(\mathcal{P},m)=\{\Lambda \subseteq \mathcal{O}_K^n | \Lambda = \pi_{\mathcal{P}}^{-1}(S), S \subseteq k_{\mathcal{P}}^n \text{ is an $m$-dimensional subspace}\}.$$

Now I know that for a fixed $m,n$ as $\mathcal{P}$ increases in norm, there is an equidistribution result commonly known as the equidistrubition of Hecke points. My question is, what is the space of lattices where these equidistribute?

Originally, I thought it would be $GL_n(K \otimes \mathbb{R})/GL_n(\mathcal{O}_K) \cdot \mathbb{R}_{>0}$. But this is the moduli space of free $\mathcal{O}_K$-modules whereas my collection $L(\mathcal{P},m)$ might have some lattices that look like $\mathcal{O}_K^{n-1}\times \mathcal{P}$. In general, all the $\mathcal{O}_K$-modules in $L(\mathcal{P},m)$ will have the same Steinitz class. 

A second candidate is the space $ \mathbb{R}_{>0} \cdot GL_n(\widehat{\mathcal{O}_K}) \backslash GL_n(\mathbb{A}_K)/GL_n(K)$ as suggested in [1]. This has as many connected components as the class group, one for each Steinitz class of $\mathcal{O}_K$-modules. But all of my points $L(\mathcal{P},m)$ will lie in a single connected component and that's not equidistribution!

So what is the right answer? Could someone please clarify this for me? Is there a reference where such subtleties have been laid out?

[1] Düzlü, Samed and Krämer, Juliane. "Application of automorphic forms to lattice problems" Journal of Mathematical Cryptology 16, no. 1 (2022): 156-197. https://doi.org/10.1515/jmc-2021-0045