Second-order ground-state energy conjecture for the Gross–Pitaevskii three-body Bose gas

Let $H_N^{\rm GP}$ be the Gross–Pitaevskii Hamiltonian, let $E_{\rm GP}(N)$ denote its ground-state energy, and let $e_{\rm GP}$ be the minimum of the corresponding Gross–Pitaevskii energy functional. Then Ground-state energy conjecture.

$$E_{\rm GP}(N)=Ne_{\rm GP}+N^{1/2}C^{(2)}_{\rm GP}+\mathcal O(1),$$

for some constant $C^{(2)}_{\rm GP}$ independent of $N$. The authors explain that an upper bound of the expected order appears accessible, but obtaining the matching lower bound and identifying the constant remain open.