There is more to it, I've just discovered.

Hilton apparently hard-codes your airline partner at the time of booking. This means that you need to get the hotel to change your preferred airline partner to LH so that the LH number shows on your bill, should you previously have had it set to a different airline or no airline at all. Alternatively retroclaim the miles afterwards.

Not everyone is having problems with this but some are. I have seen support emails sent to readers from Hilton HQ.

Also not clear if it works on reward nights ..... seems it may not.