Classification of indecomposable injective smooth semilinear representations

Let $k$ be a field, let $K=k(\Psi)$ with its standard $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action, and let $K[\binom{\Psi}{s}]$ denote the smooth $K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module associated with the $s$-element subsets of $\Psi$. Classification conjecture. For every integer $s\geq 0$, the indecomposable smooth $K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module $K[\binom{\Psi}{s}]$ is injective, and every indecomposable injective smooth $K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module is isomorphic to $K[\binom{\Psi}{s}]$ for some $s\geq 0$. This would classify all indecomposable injective smooth semilinear representations in terms of the permutation modules on finite subsets; the supplied text does not indicate whether the assertion has been proved or remains open.