I'm facing some UV unwrapping problems, and I don't know why, how or where the deformity is coming from.
Help.
Besides the UV deformation iself (which will be solved below) there is a problem in the subdivider settings. You need to set the UV Smooth option to something different from "keep boundaries".
I don't know exactly the difference between the other options, but they work well:
There is also deformity in the UV itself, as you can see in the UV editor.
This deformity is trying to minimize distortion in a 3d shape where it's just impossible to have zero distortion (because of the shape itself).
But you can edit the UV to be made of rectangles (you deliberately increase distortion at the end of the conical parts to achieve a perfect alignment).
For that, you pick one of the most central rectangles. Consider this rectangle in the steps below.
With "vertex selection", there are four steps:
Now with "face selection":
This will straighten your UV:
As an alternative to Daniel's answer, you can activate the UV Squares addon, select all in the UV Editor, open the N panel > UV Squares and click on the To Grid by Square button: