  It was shown in [S. Kaliman, M. Zaidenberg, Gromov ellipticity of cones over
projective manifolds, Math. Res. Lett. (to appear), arXiv:2303.02036 (2023)]
that the affine cones over flag manifolds and rational smooth projective
surfaces are elliptic in the sense of Gromov. The latter remains true after
successive blowups of points on these varieties. In the present note we extend
this to smooth projective spherical varieties (in particular, toric varieties)
successively blown up along smooth subvarieties. The same also holds, more
generally, for uniformly rational projective varieties, in particular, for
projective varieties covered by affine spaces. It occurs also that stably
uniformly rational complete varieties are elliptic.
