That orthographic projection is showing some pinching around the poles. (Orthographic can sort of be thought of as a view from space) You might want to try to fix that in the source map before you do anything else. In equidistant cylindrical/equirectangular the high latitudes should look "Stretched" horizontally otherwise those same areas will be "pinched when you reproject them. Try an azimuthal projection centred on each pole to get a clear look. Since its an issue of shape, Stereographic in particular would be a good choice (It's an azimuthal projection that preserves shapes, but distorts sizes).