Let $G$ be a simply-connected simlpe group over a local field $K$ and $P_x$ a maximal parahoric subgroup of it. Let $P_x^+$ be the pro-unipotent radical of $P_x$. Then $P_x/P_x^+$ is a connected reductive group over the residue field $\kappa$ with root system $\Phi_x$.
Question: Is it true that $P_x/P_x^+$ is a simply connected semisimple group with root system $\Phi_x$? If not, is it ture that $P_x/P_x^+$ has a simply connected derived group?
Any reference?