Hi assignment-brave,
I can offer only a workaround, but please feel free to ask tech support about earlier tickets.
https://www.maxon.net/en/support-center
CV4_2025_drs_25_RSfe_21.c4d
• Set up the Feather as needed but with a straight Spline, and adjust the thickness of the hair material.
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• Create a copy, and while that copy is active, press the c key
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• In the new Hair Object, set the Generate to Circle or Square.
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• With the hair Object active, press the C Key.
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• Place a material to the Hair polygon object (delete the origial one.
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• Set the Material to flat and in texture Mode, align the "frame" to the object, typically that needs a 90º rotation, and in the Object Manager> Tags> Fit to Object (Alert shows up: Include all…)
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• With that object active, object manager>Tags> Generate UVW coordinates.
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• This provides you now with a stable UV information/setup
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• To animate the Feather, a Spline Wrap is used, keep length is a good fit. I avoid using the full spline, as the endpoints are not well defined (Start/From)
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• Use a Spline as the main and another (in distance) as the Rail.
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I hope all other information is in the file; add your texture to it.
If UV is not wanted, with enough points in the Polygon results a Vertex map worked with the Spline Wrap.
As a side note, the TriPlanar is a good idea in a few cases; anything animated might not; it is, at best, a case-by-case decision.
Files: For security reasons, only Dropbox, Google, Wetransfer, Adobe, or Apple cloud services are accepted, all during uploaded with no compression and the link (as you did) pasted clearly, thanks for not shortening the URL. Sorry for the extra work. (I do not experiment with other cloud services to receive data here; we keep those out of the forum for security reasons, even if others have no concerns).
Have a great weekend.