the equilibrium density of droplets takes the form: $$\label{pzero}
  \left(\xi \frac{ \partial}{\partial \xi} -1 \right) \left( e_\mathrm{MF}(\xi) + e_\mathrm{LHY}^\mathrm{2d}(\xi)\right )  = 0.$$

We now focus on the quasi-2D regime in which $e_\mathrm{LHY}^\mathrm{2d}(\xi)$ is given by Eq. (\[e2d\]) with the last term neglected. Assuming for simplicity $g_{11}=g_{22} = 4\pi \hbar^2 a/m$, which implies $n_1=n_2 =n$, the solution of Eq. (\[pzero\]) yields: $$\label{x0}
  \xi_0 = \frac{1}{2\pi^2}  e^{  -\frac{3}{2} -  \frac{ L \delta a}{2 a^2}},$$ where we used $\delta g = 4 \pi \hbar^2 \delta a/m$. The above result leads to the following droplet density: $$\label{n2d}
  n=\frac{e^{-3/2}}{8 \pi} \frac{1}{a L^2}  e^{-\frac{L\delta a}{2a^2}}.$$

To find the conditions for a quasi-2D system, we compare the droplet density obtained with Eq. (\[e2d\]) to the one given by Eq. (\[n2d\]). We find that for $\xi
\lesssim 0.03$, the relative difference between the two results is smaller than $20\%$. We assume this condition defines the quasi-2D regime. Therefore, to have a quasi-2D system we need to have $\xi_0 \lesssim 0.03$ , and, from Eq. (\[x0\]), we find that $$\label{criterion}
  \frac{\delta a}{a} > - 4 \frac{a}{L} ,$$ According to Eq. (\[criterion\]), $\delta a$ can have arbitrary sign. Therefore, we arrive at the conclusion that droplets can be formed for 