This image shows the light fall at the edges with Sigma 8 mm. F=8
The image to the right has been corrected with an Photoshop action using 2 different selections.
No 1: 1500 pixels diameter/inverted - feather 200 pixels - levels 0-1.3-255
No 2 selection: 2500 pixels diameter/inverted - feather 150 pixels - levels 0-1.0-230
Download action

This action will only work on an image 2048 x 3072 pixels

Also the lightfall is different with different F-stops. If you stop down to 16 or 22 there is almost no lightfall.

I have not been able to get satisfactory results with one selection, and I have only tested the above action on 2 panos.
It is not perfect but it works for me.