Kantor's fiberwise formulation for rational standard conic bundles

Let $\pi:X\to S$ be a standard conic bundle, and suppose that there is a birational map

$$\Phi:X\dashrightarrow\mathbb{P}^3.$$

A Cremona transformation is a birational self-map of $\mathbb{P}^3$. Kantor's conjecture. There exists a Cremona transformation $\tau:\mathbb{P}^3\dashrightarrow\mathbb{P}^3$ such that the composition $\tau\circ\Phi:X\dashrightarrow\mathbb{P}^3$ sends a general fiber of $\pi$ either to a conic or to a line.

This is presented in the source as an equivalent formulation of Kantor's conjecture for congruences of rational curves. The supplied text gives no evidence that it has been resolved.