Let $X$ be a smooth projective variety over an algebraically closed field of characteristic zero $k$.

We call $\mathcal{P}$ the class of $k$-varieties whose objects are either:

• abelian varieties
• semi-abelian varieties (extension of abelian varieties by a torus)
• smooth hypersurfaces in large projective spaces over $k$
• smooth projective varieties of dimension $\le 3$
• products of the above

Does there exist a surjective map $Y\to X$ over $k$ with $Y$ allowed to be a blow-up of $V\in\mathcal{P}$ along a smooth closed $k$-subvariety $Z\subset V$ such that $Z\in\mathcal{P}$?