We found a palliative solution for the eyes to continue using the Foggy flag leaving the eyes better integrated.
The main problem is the Unlit mode of the material, so we change it to Default lit and then,
controlled the opacity to still have a pseudo-occlusion effect without mess with the fog.
Here is a step to step to fix it out in the master material:
Our results before the fix:
After the fix:
Hope this could be useful!