K\"ahler-Ricci flow on $\mathbf G$-spherical Fano manifolds

We prove that the Gromov-Hausdorff limit of K\"ahler-Ricci flow on a $\mathbf G$-spherical Fano manifold $X$ is a $\mathbf G$-spherical $\mathbb Q$-Fano variety $X_{\infty}$, which admits a (singular) K\"ahler-Ricci soliton. Moreover, the $\mathbf G$-spherical variety structure of $X_{\infty}$ can be constructed as a center of torus $\mathbb C^*$-degeneration of $X$ induced by an element in the Lie algebra of Cartan torus of $\mathbf G$.