Gross–Pitaevskii limit conjecture for combined two- and three-body interactions

Let $H_N^{\rm Comb}$ be the Hamiltonian with two-body potential $W$ at scaling parameter $\beta=1$ and three-body potential $V$. Let $e^{\rm Comb}_{\rm GP}(b_1,b_2)$ be the minimum of the combined Gross–Pitaevskii functional, let $b(W)$ be the scattering energy of $W$, and let $b_{\mathcal M}(V)$ be the three-body scattering parameter. Then Combined-interaction Gross–Pitaevskii conjecture.

$$\lim_{N\to\infty}\frac{\inf\sigma(H_N^{\rm Comb})}{N}=e^{\rm Comb}_{\rm GP}\left(b(W),b_{\mathcal M}(V)\right).$$

The case $\beta=1$ is challenging because both interactions are critically scaled, so correlations at both the two- and three-body levels contribute to the leading order. The limit remains open.