Determine the exact minimum at non-distinguished lengths

Determine the exact value of J(L), the infimum of the mean geodesic distance over simple closed curves of prescribed length L, for every L>2π not belonging to the set {L_n=2π/sin(π/(2n)): n∈N}.

Background

The paper establishes exact values of J(L) at the distinguished lengths L_n and gives the lower bound j(L)≤J(L) for arbitrary L>2π. Monotonicity of J supplies upper bounds from the preceding distinguished length, but these bounds do not determine J(L) between successive L_n.

The authors explicitly leave unresolved the exact minimization problem for all remaining lengths greater than 2π. The issue is tied to whether there exist sphere-filling curves with the required covering radius and collar structure at radii that are not among the discrete values π/(2n).