Algebraic ellipticity of quartic double spaces

We prove that every smooth quartic double $n$-fold, $n\ge 2$, over an algebraically closed field of characteristic zero is algebraically elliptic in Gromov’s sense. In particular, every smooth quartic double solid is algebraically elliptic, which answers Question~4.28 in [M. Zaidenberg, Algebraic Gromov ellipticity: a brief survey, Taiwanese J. Math. 29 (2025), no. 6, 1681-1705]. We also prove that removing any closed subset of codimension at least two preserves algebraic ellipticity for smooth quartic double spaces and smooth cubic hypersurfaces of dimension at least two.