Kantor's conjecture on index-one congruences of rational curves

A congruence of curves in $\mathbb{P}^3$ is an irreducible two-dimensional family of curves covering $\mathbb{P}^3$; its index is the number of curves through a general point. Kantor's conjecture. For any index-one congruence $\mathcal C$ of rational curves in $\mathbb{P}^3$, there exists a Cremona transformation

$$\tau:\mathbb{P}^3\dashrightarrow\mathbb{P}^3$$

that sends $\mathcal C$ to a two-dimensional family of conics or lines.

The conjecture is a classical birational-geometric formulation equivalent in the paper to the relevant conic-bundle rationality conjecture. No resolution is supplied in the source.