Lieb–Thirring conjecture for all particle numbers

Prove that the optimal constants K_N in the one-dimensional kinetic Lieb–Thirring inequalities satisfy K_N=π²/4 for every positive integer N.

Background

For orthonormal functions u₁,…,u_N in H¹(R;C), the paper defines K_N as the optimal constant in the inequality relating the total kinetic energy to the cube of the density ρ=Σ|u_j|². Keller's sharp one-bound-state result gives K₁=π²/4.

The paper states the all-N conjecture but proves only a two-state result, namely K₂=π²/4, together with the corresponding bound for two negative eigenvalues. Thus the general assertion for every N remains unresolved in the context presented.