The no-conjugate-points conjecture for the Zermelo navigation problem with a vortex

Let $M$ be the domain of the Zermelo navigation problem, let $x_0\in M$ be an initial point, and let $\mu\in\mathbb{R}^{*}$. The conjugate locus is the set of endpoints that are conjugate to $x_0$ along geodesics.

No-conjugate-points conjecture. For any $(x_0,\mu)\in M\times\mathbb{R}^{*}$, the conjugate locus is empty.

The conjecture asserts the absence of conjugate points for every initial point and every nonzero vortex parameter, extending the numerical observations for weak and strong drift. Its status is not resolved in the supplied source.