Set your white walls at 210 and your texture multiplier at 0.8. So you should get the natural color bleeding. If this isn’t enough you need a special material (it’s new and I don’t used it yet). Also it looks like your rendering is over exposured. Adjust the brightness until the wall are light grey looking.
A very simple solution would be to place a copy of your surfaces below and assign a grey color.
Edit: try the “override” material and you can control the color of the GI