The 58-line bound conjecture for K3 quartic surfaces in characteristic 3

Let $X$ be a K3 quartic surface defined over a field of characteristic $3$. The 58-line bound conjecture. The surface $X$ contains $112$ lines or at most $58$ lines. The theorem preceding this conjecture proves the weaker bound of $67$ lines outside the $112$-line Fermat case; the conjecture would sharpen that bound to $58$, and the paper provides evidence from configurations of lines on smooth quartic surfaces.