Exact formula for the minimal length of a geodesic with at least k self-intersections

Let $M_k$ be the infimum of lengths of geodesics with self-intersection number at least $k$ among all finite-type hyperbolic surfaces. A minimal-length geodesic conjecture asserts that, for $k\geqslant 1$,

$$M_k=2\cosh^{-1}(1+2k)=2\log(1+2k+2\sqrt{k^2+k}).$$

Moreover, equality is attained when $\Gamma$ is a corkscrew geodesic on a thrice-punctured sphere. This refines Basmajian's logarithmic bounds for $M_k$ by proposing an exact value and an explicit extremal surface and geodesic; the supplied text does not indicate whether the claim has been proved or remains open.