Phase interference and comb filtering dominate the challenges we face. Reflections are the biggest challenge as they arrive late and cause constructive and destructive interference.
What is important is not phase at 10kHz, but phase relative to the woofer at and below the crossover point. Figure the wavelength at 2kHz is about 6". This is enough to make phase matter. With a 5kHz crossover point and a wavelength of .225", phase means much less since minute head movements will negate the issue.
One thing to try before adjusting phase or crossover point is to move around. If you hear an issue, move your head a foot and listen.
Diffraction matters, and this is well documented. Tweeter mounting and position relative to other objects is very important. That is why spheres work so well.
I have spent a decent amount of time with a variety of 3-way systems in several orientations, the H701 allows for experimentation from the drivers seat. Tweeter phase is relative to the rest of the system. Can it make a difference? Of course. There are so many complex interactions, every change matters. Does the magnitude of the change vary? Of course. Is it a very easy thing to experiment with? Definitely.