Conjecture on the geodesic-covering number of standard regular surfaces

Let $Y_g$ be a standard regular hyperbolic surface of genus $g\geq 2$, and let $K_{Y_g}$ denote its geodesic-covering number. Geodesic-covering conjecture for standard regular surfaces. One has

$$K_{Y_g}\lesssim g.$$

The conjecture would improve the paper's bounds for distinct distances on standard regular surfaces by sharpening the available geodesic-covering estimate; the source presents it as an approach to improving the factor depending on the genus.