For meshes like this with mostly hexagonal faces combined with negative Gaussian curvature, planarization is not easy at all, and it isn’t generally possible simply by adding planarization goals.
That approach can work well (with just a little bit of smoothing or other goals to keep the faces from distorting too much) for quad meshes, or hex meshes with all positive curvature, since the points often don’t have to move too far across the surface to make the faces planar.
With negative curvature and hexagons though, some of the faces need to become non-convex, and this doesn’t happen spontaneously during relaxation. There was a thread about this recently here: