Unable to export skybox images without gradient banding

I created a skybox by creating a cube and then assigning a material/texture to it with an Env Map type and then used “save envmap” to export the six sides (followed some tutorial I googled). The problem is that I have a sphere that surrounds this cube and has all the faces turned inwards. The texture on this sphere is a Clouds texture with Blender Original noise biase and a typical colour gradient. This Blender Clouds texture always shows up as 256 colours (8bit) in the exported env maps and the gradient banding is really bad.

What do I need to do to get the Clouds texture to use higher bit depth colours, or is the problem with exporting via “save envmap”? Here’s a picture of my skybox with the banding (most obvious near the planet):


here’s another: