I don't know the specific type, just that its a PTO pump and not a power beyond from the JD. I usually run it around 10,000 - 12,000 RPM, slightly above idle.
That may be an issue right there. PTO pumps only deliver rated output at rated RPM, in this case 540 RPM on the PTO shaft. To get 540 on the PTO shaft, you need to have the engine RPM up near the PTO rateing mark on the tach. In the case of a diesel tractor, that mark is probably 2300-2400 RPM.
in theory, the pump should be able to deliver full pressure at reduced flow at a lower RPM, but in reality, it dosn't always work like that, as at lower RPM, the speed of the internal pump impellar can allow for more backflow/cavitation...
The input check valve should not allow any reverse flow back toward the pump though, and if the pump is not able to make enough pressure to lift the boom, at the most it should just stay in one place, and not go down. It going down when commanded up still points to fluid leaking by the relief valve.
A quick check to try would be to run the tractor engine up to rated PTO RPM on the tach, and see how the BH performs...
Do you know what fluid is in the BH? If it is still the chinese stuff, I would highly reccomend getting rid of it, flushing the system with kerosene and putting in AW-32 hydraulic fluid. The folks I got my BH from said the only real problems they ever had with these hoes were caused by crap in the oil interfering with spool and relief valve function.