Drinfeld has constructed two towers of covers $(\mathcal{M}^n_{LT})_n$ and $(\mathcal{M}^n_{Dr})_n$ as deformation spaces of formal modules. It is known that the supercuspidal part of the geometric étale $\ell$-adique cohomology with compact support of these spaces provides geometric realizations of local Langlands and Jacquet-Langlands correspondances. We would like to establish the same correspondances for De Rham cohomology with compact support. We will explain how we could hope for a proof of these result in the particular case of the first cover of the Lubin-Tate by adapting the methods in Yoshida's thesis he exhibits a link between the geometry of these spaces and Deligne-Lusztig variety. It relies on the generalisation of an excision theorem of Grosse-Klönne, already used to prove the analogus result on the Drinfeld side.