My question is related to another one I read here in Overflow. I am reading Scholze's papers about moduli spaces of $p$-divisible groups and elliptic curves, and I am very interested in the formal geometry involved there. Actually, I noticed that there is a paper by Andreatta, Iovita and Pilloni, titled Le halo spectral, which seems to deal with formal integral models of Scholze's towers.

First, if I well understand Scholze, talking about elliptic curves, there is a perfectoid space $\mathcal{X}_{\infty}(\epsilon)$ which gives the "tilda limit" of modular curves $\mathcal{X}_{\Gamma(p^n)}(\epsilon)$ where each $\mathcal{X}_{\Gamma(p^n)}(\epsilon)$ describes open neighborhoods of the ordinary locus of the $\Gamma_1(N)$ modular curve, where the universal elliptic curve coming from pullback is not too supersingular. Actually, the construction of this object is performed by computing the adic generic fiber of the formal scheme $\mathfrak{X}_{\infty}(\epsilon)$ which is the real limit (in the category of formal schemes) of integral models of $\mathcal{X}_{\Gamma_(p^n)}(\epsilon)$ where the maps in the inverse system are given by a lifting of mod $p$-Frobenius.

A very similar construction is performed in chapter $6$ of Andreatta, Iovita and Pilloni's paper, where they construct the integral anticanonical tower $\mathfrak{X}_{\infty}$ exactly in the same way, but working over a basis which is a suitable blowup of an integral model of Coleman's weight space. Now, I'm just wondering whether or not it is possible to interpret these "infinite" level spaces as moduli spaces of elliptic curves plus a new kind of level structure. Somewhere in Scholze's paper it is mentioned that a point of $\mathcal{X}_{\infty}$ over $\text{Spa}(C,\mathcal{O}_C)$, where $C$ is a complete algebraically closed extension of $\mathbb{Q}_p$ corresponds to an elliptic curve over $C$ with a trivialization of its Tate module. Now, why is this true? It's not mentioned in Scholze and I cannot prove it. Moreover, does a similar description hold for different kind of points, e.g. $\text{Spa}(R,R^+)$ with $R$ a perfectoid $\mathbb{Q}_p$-algebra? Moreover, does the same intepretation hold for its formal integral model? And what about the Andreatta, Iovita and Pilloni's tower? Is it true that it parametrizes elliptic curves with $p$-divisible groups playing the role of the canonical subgroup? The point essentially is, does this object gives by pullback a universal elliptic curve? Which kind of level does it have a similar elliptic curve?