Let \(n\geq1\) be an integer, and \(F\) be a non-Archimedean local field whose ring of integers is \(O_{F}\) and residue field \(\mathbb F_{q}\). Suppose \(H_0\) is a one-dimensional formal \(O_{F}\)-module over \(\mathbb F_{q}\) of height \(n\), and let \(M_{H_0,m}\) is the rigid analytic space parametrizing deformations of \(H_0\) together with a Drinfeld level \(m\) structure. See [M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)]. The local Langlands correspondence and the Jacquet-Langlands correspondence for \(\mathrm{GL}_{n}(K)\) are both realized in the \(l\)-adic cohomology of the Lubin-Tate tower \(M_{H_0,\infty}=\lim M_{H_0,m}\). At the present, there is only a global proof for this fact in the literature. The second author in [Doc. Math., J. DMV 15, 981–1007 (2010; Zbl 1211.14027)] and [“Semistable model for modular curves of arbitrary level”, Preprint, arXiv:1010.4241] attempts to obtain a local proof the fact by constructing a nice model of \(M_{H_0,m}\). The original idea goes back to the work of T. Yoshida who was able to construct a formal model for \(M_{H_0,1}\) [Adv. Stud. Pure Math. 58, 361–402 (2010; Zbl 1257.11103)]. Yoshida showed by purely local methods that the local Langlands correspondence for depth zero supercuspidal representations of \(\mathrm{GL}_{n}(K)\) is realized in the cohomology of \(M_{H_0,1}\). The authors of the paper under review use this idea to generalize the results to large class of supercuspidals of positive depth. They work directly with \(M_{H_0,\infty}\) rather than any particular \(M_{H_0,m}\) of the tower (see [loc. cit. arXiv:1010.4241] and [P. Scholze and the second author, Camb. J. Math. 1, No. 2, 145–237 (2013; Zbl 1349.14149)]).