Hi @ModernArtist!
You’re applying the strands to a convex surface, so some degree of flare at the edges is to be expected. If you’ll notice, in that video the user is projecting the strands from a flat surface. This is physically realistic.
The relevant modifiers in the Fibermesh controls are:
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Gravity. You may have to max this out, and may need to adjust the curve profile as well.
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Segments–add more of them. The more points there are on the strands, the more responsive they will be.
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Revolve/Twist–disable. Any setting that adds a twist or a curve will complicate the strand direction, so reduce all these settings to zero while trying to get the direction as straight as possible.
Here are some strands with Max Gravity and additional segments:

Also, don’t forget about grooming. As long as you are applying the strands to a curved surface, the strands will flare out in the direction of the surface normals to some extent. However, Fibermesh strands are designed to be used with the Groom type brushes to style the strands once they are converted into a subtool. It was a simple matter to comb out these strands after they were created:
