Second-order low-density expansion conjecture for the dilute three-body Bose gas

Let $e_{\rm 3B}$ denote the ground-state energy per unit volume in the thermodynamic limit, let $\rho$ be the particle density, and let $b_{\mathcal M}(V)$ be the three-body scattering parameter. For fixed interaction potential $V$, consider the low-density limit $\rho\to0$. Then Low-density expansion conjecture.

$$e_{\rm 3B}=\frac{1}{6}\rho^3b_{\mathcal M}(V)\left(1+C^{(2)}_{\rm TL}(V)\rho+o(\rho)\right),$$

with a constant $C^{(2)}_{\rm TL}(V)\in\mathbb R$. This predicts a second-order correction to the leading dilute three-body energy and raises the question of whether the correction is universal; its validity and the nature of the coefficient remain open.