Homotopy $4$-spheres associated to an infinite order loose cork

We show the homotopy spheres $\Sigma_{n} = -W\smile_{f^{n}}W$, formed by doubling the infinite order loose-cork $(W,f)$ by iterates of the cork diffeomorphism $f: \partial W \to \partial W$ is $S^4$. To do this we first show that $\Sigma_{n}$ are obtained by Gluck twistings of $S^4$; then from this we show how to cancel $3$-handles of $\Sigma_{n}$ and identify it by $S^{4}$.