  In this note we search for the ground state, in infinite volume, of the $D=3$
Wilson-Fisher conformal $O(4)$ model, at nonzero values of the two independent
charge densities $\rho_{1,2}$. Using an effective theory valid on scales longer
than the scale defined by the charge density, we show that the ground-state
configuration is inhomogeneous for generic ratios $\rho_1 / \rho_2$. This
result confirms, within the context of a well-defined effective theory, a
recent no-go result of Alvarez-Gaume' et al. We also show that any spatially
periodic ground state solutions have an energetic preference towards longer
periods, within some range of $\rho_1 / \rho_2$ containing a neighborhood of
zero. This suggests that the scale of variation of the ground state solution in
finite volume will be the infrared scale, and that the use of the effective
theory at large charge in finite volume is self-consistent.
