2n-Weak module amenability of semigroup algebras

Let $S$ be an inverse semigroup with the set of idempotents $E$. We prove that the semigroup algebra ${\mathrm{\ell }}^1(S)$ is always $2n$-weakly module amenable as an ${\mathrm{\ell }}^1(E)$-module, for any $n\in \mathbb{N}$, where $E$ acts on $S$ trivially from the left and by multiplication from the right.