I’ve tried about 2 days to wrap this image on something like a sphere, but without success.
I always get bad uv mapping on the top and bottom of the sphere. I’ve tried round cube, uv sphere, octahedron sphere and a lot of uv mapping variations. Nothing helped.
This image shows a round cube with sphere projection. Direction: Align to Object
As you can see your texure is very distorted on the poles. so the problem is with the texture not with the UV or mesh. Do you need to use exactly this texture without any modifications? Because I would map different map (can be texture painted) on the pole areas