Double Gaussian half-plane conjecture in the plane

Let $f(x,y)=e^{\psi(x,y)}$ be the normalized sum of two Gaussian densities with the same variance and different centers. The line connecting the two centers determines a perpendicular direction. Double Gaussian half-plane conjecture. Isoperimetric regions are half-planes enclosed by lines perpendicular to the line connecting the two centers.

This is the planar double Gaussian case of the conjecture attributed to Cañete et al. The paper describes it as evidence for the claim, proving that horizontal and vertical lines are the only stationary lines and that vertical lines outperform horizontal lines, but does not establish the full assertion.