Rigidity of the subelliptic heat kernel on $\mathrm{SU}(2)$

We study heat kernel rigidity for the Lie group $\mathrm{SU}\left(2\right)$ kernel equipped with a sub-Riemannian structure. We prove that a metric measure space equipped with a heat kernel of a special form is bundle-isometric to the Hopf fibration $\mathrm{U}\left(1\right)\to \mathrm{SU}\left(2\right)\to {\mathbb{C}\mathbb{P}}^1$, which coincides with the sub-Riemannian sphere $\mathrm{SU}\left(2\right)$.