Ok, here is another gradient mesh, which renders much better.
General commetns on drawing it with linear and radial gradients:
use the snapping tool to snap the gradient handles to the path nodes,
set height/width/node position precisely so that your image makes up a square and the corners right.
If you want the gradient to have transparency, AND avoid the anti-aliasing rendering gap issue, the only way of achieving that is by using filters which is a bit of a bummer.
For filling up the gaps I used overlapping, hence the corners are not quarter sectors and the sides are not rectangles either. Check it in outlines only rendering mode.
Once you draw a black to white gradient with 0 transparency, with the filter you can map in other colours, just as in the file.
Only problem with that is it's limited to 256 steps since that's the whole luminance range used for the input -a colormatrix filter primitive is used in luminance to alpha mode-.
Would need some tricks avoiding banding with more than 2 colour fades.
And in general rendering the filter may be a
bit off since 0.91 and gradients may not be 100% accurate.
(An "extreme" example
here. Very small steps in a gradient are not rendered with a smooth fade that aligns fine.)