I could only add that Anton Kumaigorodsky tried to develop private channels routing for his IMMORTAN library but some threat vectors which may realize with such routing stopped him and as he said, it may require features not available yet. For example Eltoo.

notgeld

bitcoin

How are private channels included in LN paths?

pillar

You can't gossip about a channel that isn't yours. The gossip has to be signed by both nodes making up the channel to be accepted and relayed. So both sides must agree to announce it.

Here's the proof of concept: https://github.com/BitcoinDevShop/hidden-lightning-network
