Absence of binding in one dimension for the nonlinear Schrödinger equation

Let $d=1$, let $N\geq2$, and let $J(N)$ be the constrained ground-state energy for $N$ orthonormal functions, with exponent $p$ in the range $2\leq p<3$.

Absence-of-binding conjecture. For all $2\leq p<3$ and all $N\geq2$, $J(N)$ has no minimiser and

$$J(N)=N J(1).$$

This would mean that in this one-dimensional range the energy of $N$ orthonormal particles is exactly the sum of the one-particle energies, so binding never occurs. The paper states this as a probability rather than a proved result, and no resolution is supplied here.