That's the Spilhaus equal-area in polar aspect with five lobes and "CON" (basically lobe pointiness) set to Spilhaus's preferred value of 1.145. Pretty good apart from Africa and South America, which can again be improved by replacing the azimuthal equal-area part with an equal-area conic projection:
although Greenland is now rather elongated.
For the purposes of a flat-earth projection, I agree that this would probably be better, but I've kept the interruption down the Bering Strait for now so that I don't have to make a little tab for north-eastern Greenland when the interruption runs west of Africa.
You can also make a very similar projection by replacing the equidistant conic part of the Bartholomew regional with an equal-area conic projection:
I still prefer the version with the equidistant conic, though.