Algebraic Gromov's ellipticity of cubic hypersurfaces

We show that every smooth cubic hypersurface $X$ in $\mathbb{P}^{n+1}$, $n\ge 2$ is algebraically elliptic in Gromov's sense. This gives the first examples of non-rational projective manifolds elliptic in Gromov's sense. We also deduce that the punctured affine cone over $X$ is elliptic.