Extremal lightness of high weighted-girth graphs

Determine the asymptotic value of the extremal lightness function $\lambda(n,k)$, defined as the supremum of the classical lightness of n-node weighted graphs with weighted girth greater than k.

Background

The function $\lambda(n,k)$ governs the best possible lightness of ordinary graph spanners through the weighted-girth framework of Elkin, Neiman, and Solomon. The paper uses the currently known upper and lower bounds on this function to derive corresponding bounds for fault-tolerant spanners, but the exact asymptotic behavior of $\lambda$ remains unresolved.