Universal elliptic curve over anticanonical tower

While I'm reading Scholze's paper "On torsion in the cohomology of locally symmetric varieties" he constructs the anticanonical tower passing through the construction of an integral model $$X_{\infty}$$ over $$\text{Spf}\mathbb{Z}_p^{\text{cycl}}$$ which is the limit along Frobenius of integral models of strict neighborhoods of the ordinary locus. Over every neighborhood we have a universal elliptic curve given by base change, as they are defined by blow up. Is there also a universal elliptic curve over the limit? How is it constructed? Is it the base change of one of the universal elliptic curve at finite level?