Extended Langer–Singer lower-bound conjecture for long null-homotopic curves

Let $N$ be a Riemannian manifold with sectional curvature at most $\delta<0$, and let $\gamma$ be a null-homotopic regular closed curve in $N$ whose length is greater than $2\pi/\sqrt{-\delta}$. Extended Langer–Singer conjecture. The total squared curvature satisfies

$$\int_\gamma \kappa^2\,ds\ge 4\pi\sqrt{-\delta}.$$

This is the long-curve case of the reformulation proposed in the paper; the text gives partial bounds and does not state a general resolution.