  In this paper we disprove part of a conjecture of Lieb and Thirring
concerning the best constant in their eponymous inequality. We prove that the
best Lieb-Thirring constant when the eigenvalues of a Schr\"odinger operator
$-\Delta+V(x)$ are raised to the power $\kappa$ is never given by the one-bound
state case when $\kappa>\max(0,2-d/2)$ in space dimension $d\geq1$. When in
addition $\kappa\geq1$ we prove that this best constant is never attained for a
potential having finitely many eigenvalues. The method to obtain the first
result is to carefully compute the exponentially small interaction between two
Gagliardo-Nirenberg optimisers placed far away. For the second result, we study
the dual version of the Lieb-Thirring inequality, in the same spirit as in Part
I of this work (D. Gontier, M. Lewin & F.Q. Nazar, arXiv:2002.04963). In a
different but related direction, we also show that the cubic nonlinear
Schr\"odinger equation admits no orthonormal ground state in 1D, for more than
one function.
