Edit: sorry for such a long post, but I wanted to give as much info as possible.
Thanks for your response. Just baking the lighting (without lightprobes) didn't change anything for me, but with Light Probes did.
My hope was to use Ray-Traced Global Illumination at render time and SSGI during development time as it is faster, and the
docs state "It is a more accurate alternative to Screen Space Global Illumination, Light Probes and lightmaps." This gave me the impression that it is a replacement for all of these, but even with RTXGI the hair and eyes glow in the dark (before baking lighting with light probes). One of my reasons for using RTXGI was to hopefully avoid the hassle of baking lighting (and bad results) of light probes.
After baking lighting though, another problem arises with the eyes, the white part of the eyes becomes black as seen in the screenshots below.
All screenshots are after baking the lighting with light probes and with the local reflection probes enabled.
The final 3 screenshots are a comparison between the default auto-setup on the left and Victor.Soupday auto setup on the right. The default auto-setup uses HDRP/Lit for the hair and eyes.
I upped the brightness for all of them to make them easier to see.
Screenshot 1 with RTXGI enabled and the eye occluders active
:
Screenshot 2 with RTXGI enabled and the eye occluders removed:

Screenshot 3 with SSGI disabled completely and with the eye occluders removed:

Screenshot 3 with the brightness turned up (I guess this seems normal):

Screenshot 2 with the brightness turned up. (to properly show how black the eyes are compared to the surroundings):

Screenshot 6 with SSGI (ray-marching not ray-tracing) and I guess that looks normal. So maybe the problem here is just with ray-tracing:

Comparison:
Screenshot 7 with SSGI:

Screenshot 8 with RTXGI:

Screenshot 9 with SSGI disabled completely: