Minimal condition number conjecture for lattices

Let $R_{g,\Lambda}$ denote the matrix associated with the Gaussian $g$ and a lattice $\Lambda$, and let ${x_1}(\Lambda)$ denote the lattice parameter used in the source. Minimal condition number conjecture.

$$R_{g,\Lambda}=\sqrt[3]{2}$$

is the minimal condition number achievable for any lattice $\Lambda$ with ${x_1}(\Lambda)=1/2$. The claim is motivated by a numerical condition number for the hexagonal lattice; no proof or resolution is supplied.