Asymptotic estimate for the interaction parameter in optical-lattice condensates

Let $\widehat{\tau}$, $\widehat{g}$, and $\tau$ denote the parameters introduced in the model, and let $\epsilon$ tend to zero. Asymptotic interaction-parameter conjecture.

$$\widehat{\tau}=\widehat{g}\,o(\tau),\qquad \text{as }\epsilon\rightarrow 0.$$

This is presented as a reasonable heuristic guess, implicitly used by physicists, for the relative asymptotic size of the parameters in the Bose–Einstein condensate model. The supplied text gives no proof or resolution, so its status remains open.