How to combine WebGPU SSGI + Fog

Hi I have a three.js application using the WebGPU renderer and I have a scene with quite a large floor. To make the floor blend into the background color i wanted to use a distance fog. However, since I am also using the SSGI post-processing as in three.js examples , and more specifically the ambient occlusion, this makes my floor significantly darker than the background because it is just using the depth texture.

I did think about doing my fog as a post-processing effect as well (calculating it manually depending on the scene depth from the camera) but this has the side effect of not working with transmissive materials for example.

Using both of these solutions kind of works but adapting the color is quite a bit more difficult and it just feels a bit clunky overall so I wanted to ask: do you guys have a better idea on how to solve this?