Prime Fano $4$-folds with semi-free torus actions

Let $X$ be a smooth complex prime Fano fourfold having a semi-free action of ${\mathbb{C}}^{\ast }$, then $X$ is contained in one of the families ${\mathbb{P}}^4,Q^4,W_5,X_8^m$. All of the families contain members that have a semi-free ${\mathbb{C}}^{\ast }$-action.