A1-cylinders in smooth rational projective varieties

Determine whether every smooth rational projective variety contains an open subset isomorphic to the product of the affine line with a variety.

Background

An A1-cylinder is an open subset of a variety isomorphic to A1 times another variety. The paper distinguishes A1-ruled varieties, which contain such cylinders, from merely ruled varieties, which are only birational to such products. Although an A1-ruled variety is ruled, the converse is not known in general, even for smooth rational projective varieties.