Does it have a warranty ? I see it's quite expensive and out of my leauage.
But I'd open it up, and see what the motherboard is doing. Check the PSU section too. A lot of it is probably like a mini-PC, and has a strict startup routine. What happens when ram and rom go wrong.
I'd want to know if just the screen locks up, would you normally hear changes after pressing buttons, like relays?
There's probably some microcontroller or other such chip, that should be sensing this problem, and you would think, would want to time out and reset. Does it do a RAM check and just get stuck and never finish? Does some circuit or chip somewhere not initialize, and rest of it sits there waiting forever ?
Here's a service manual, there's a lot pf troubleshooting sections in there, if you want to open up the HP.