De Rham cohomology of Lubin-Tate spaces and Drinfeld spaces

Let $d\ge 1$ be an integer. We use the methods introduced by Lue Pan to prove that the compactly supported cohomology of Lubin-Tate towers and Drinfeld towers are isomorphic, as ${\text{GL}}_{d+1}(L)\times D_{L,\frac{1}{d+1}}^{\times }$-modules.