I do not know for LND but I heard that core lightning do it by default in the sense that it try to obfuscate the transactions also using shadows nodes that are not on the path to the receiver. Doing that has of course some disadvantages one of which is probability of higher fees due to longer path and more failed payments. ;-)
Do you have a reference for that? I never heard of shadow nodes so far.
A solution to this problem is to route all your LN transactions through a public node that you secretly control so that you can be sure that it will never cooperate with other nodes to harm your privacy as a sender.
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that's because it was shadow route :-) you can find an explanation here; https://voltage.cloud/blog/bitcoin-lightning-network/what-are-the-differences-between-lnd-and-cln/
under privacy.
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Thanks! I will try out core lightning
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