Kottwitz has built canonical integral models for a large family of PEL Shimura varieties, associated to a certain reductive group $G$ over $\mathbb Q$, when the structure level has the form $K = K_pK^p$ with $K^p \subset G(\mathbb A_f^p)$ small enough and $K_p \subset G(\mathbb Q_p)$ hyperspecial. This construction admits a moduli interpretation, classifying abelian schemes with integral additional structures.
Using the notion of full set of sections, Mantovan has built integral models for the same Shimura varieties with structure level of the form $K_m = K_{p,m}K^p$ where, for $m\geq 0$, we have
$$K_{p,m} := \{g \in K_p \,|\, g \cong 1 \mod p^m\Lambda \}.$$
To explain the notations, the group $G$ arises as the group of symplectic similitudes of some space $V$ equipped with an alternate form. One hypothesis to ensure the existence of hyperspecial $K_p$ is that $V\otimes_{\mathbb Q} \mathbb Q_p$ must contain a self-dual lattice $\Lambda$, so that we can take $K_p = \mathrm{Stab}_{G}(\Lambda)$. Note that $K_p = K_{p,0}$.
The integral models defined by Mantovan also admits a moduli interpretation, refining the problem considered by Kottwitz.
Nowadays, for more general levels $K = K_{\mathrm{par}}K^p$ where $K_{\mathrm{par}} \subset G(\mathbb Q_p)$ is parahoric, we can consider the canonical integral model built by Kisin and Pappas. Their construction works for Shimura varieties of abelian type, a much broader class than those of PEL type, and as such Kisin and Pappas did not rely on moduli problems to build their integral models.
Are there known cases of PEL Shimura variety with general parahoric level structure, whose integral model built by Kisin and Pappas admits a moduli interpretation ? As a rule of thumb, is a moduli interpretation even expected to exist at such parahoric level structure ?
I am not aware of any such case in the litterature, but if it does exist I'd be very interested in learning more about it.
