Put some relief in the neck, or the action is too low. My bet is on the neck not having any relief. Capo the first fret, touch the last fret. Somewhere around the 7th-8th fret check with a feeler gauge. Should be able to get a .010" between the top of the fret and the bottom of the string. If you can't, adjust the truss rod until you can. Let the neck sit for a little bit to stabilize and then check it again. Once it's good, set your action keeping with the radius you have at the bridge. It takes some tweaking but well worth it once set.
And yes, there should be a little bit of fallaway at the higher frets like described and shown in the video you referenced.
Actually, more relief is going to exacerbate the problem. Falloff in the upper frets will help.
Relief creates a concave fretboard, falloff essentially creates a convex fretboard. By tightening the trussrod further you create the same convexity as filing down the upper frets (falloff, or fallaway).