This unfortunately did not work.
For the others in the future: The final solution for my case was to change the motherboard (it was - luckily from my point of view - in warranty) and the HP support was really good.
Simply flashing it again was impossible, as I encountered an error message (I don't recall literaly - it said sth. like Intel chipsets are not supported). I made a couple of furthermore trials (online / offline update, some variants of Update USB Sticks / DVDs), all without luck.
Cheers,
André