I'm testing on 1.20.1 latest forge which is 47.3.22
When I use a mod called Toni's Immersive Lanterns and Customizable Player Models with a 32x skin it places the lantern on my head for some reason.
It took ages to find the problem but yeah not sure if this is an issue with Immersive Lanterns, Accessories, or CPM
I can't say I've had this issue with Curious Lanterns since the placement worked fine on there with CPM installed

CPM is not currently supported and is planned but not currently possible without more investigation.

More investigation into Curios Laterns seems to indicate they use general matrix translations rather than using the given model part meaning that that they are always within the same location regardless of any of the model part positions leading to issues where the translation will not follow the model parts in many cases.

Accessories attempt to translate to the given model parts, which can be a problem when mods adjust/remove targeted parts, leading to improper translations.

Overall currently CPM or Figura will have Accessories render in incorrect position (EMF is supported though some hacky work around though) until some API is made to have the Accessory Renders model targets be retargeted to proper parts or adjust general transforms some how.