Hyperbolic surfaces with sublinearly many systoles that fill

For any $\epsilon >0$, we construct a closed hyperbolic surface of genus $g=g(\epsilon )$ with a set of at most $\epsilon g$ systoles that fill, meaning that each component of the complement of their union is contractible. This surface is also a critical point of index at most $\epsilon g$ for the systole function, disproving the lower bound of $2g-1$ posited by Schmutz Schaller.