Inkscape

Nice!

256 levels isn't very many (but it is what we are unfortunately limited to in SVG for the moment). The lines you are seeing are enhanced due to Mach Banding (see the Wikipedia article for details). It's interesting that the banding is most obvious in the yellow to orange section and in the blue section. Might be a factor of luminosity. To deal with the banding, the gradients need to be dithered. For that we need support inside of Cairo. (Essentially one renders the gradient at a higher bitmap depth and then quasi-randomly adds noise before reducing down to 8 bits per channel.)

I took your file and replaced the filter by directly coloring the mesh gradients. I see no banding when I do that.

Thank you!

That limitation doesn't sound so good. The concept of this gradient map filter was to make the gradient mesh work something similar to linear gradients -you could switch from one definition to another, while keeping the handles/mesh nodes intact.
And that it would work as how "bumpmaps" are used on a texture of a model with lower polygon count.