I would consider all these suggestions, as to what might be the issue, However a fresh rebuilt pump for the factory shouldn't have the issues of a well worn pump....
Yes, you would think a rebuilt pump would be tested, etc. but HH piston is not rebuildable, yet.
To see what I mean take all injector feed tubes off of HH and turn motor over. If fuel is not shooting
over a foot high out of holes in HH, start up speed fuel pressure is too low to get fuel to open injectors.
TM's explain necessary fuel pressure to open injectors.
Also, you might see if you are getting a back-flow of fuel from HH when you turn motor off. I changed
my fuel lines to a clear nylon type so I could watch for this and look for air leaks, too.
If you still have doubts about HH pressure maybe you can find someone who has the equipment
to give a good pressure reading at startup speed.
Good luck.