The equality of planarity and uniruledness invariants for affine varieties

Let $V$ be a $k$-uniruled affine variety. The invariants $\operatorname{P}(V)$, $\operatorname{U}(V)$, and $\operatorname{AU}(V)$ denote the planarity, uniruledness, and algebraic uniruledness invariants, respectively.

The affine uniruledness conjecture.

$$\operatorname{P}(V)<\infty,\qquad \operatorname{P}(V)=\operatorname{U}(V)=\operatorname{AU}(V).$$

The conjecture proposes that $k$-uniruledness forces finite planarity and identifies the three invariants. The paper motivates it through results for hypersurface complements and through the relation between holomorphic curves and the contact invariants of their boundaries.