Can we avoid third party trust for mobile / desktop wallets with respect to lightning network channels? Mobile and desktop wallets are nodes that are not connected 24/7 (when the application is closed, computer is powered off, etc...).

The fact that they are not connected 24/7 creates mainly one problem: if a fraudulent node closes a channel with you and sends an outdated CSV transaction, you won't see it and can't invalidate it in time if you don't launch your wallet periodicaly.

A solution to this problem is to send all your CSV transactions to a service (connected 24/7 to the blockchain) and trust him enough to broadcast theses transactions in case a fraudulent node decides to close one of your channels.

And you must really trust this service because if he is affiliated with the fraudulent node you either need to send him a higher fee than the value of the transaction or he will prefer to keep the fraudulent transaction.

The conclusion is that you can't trust such a service.

Is there an other solution to avoid having to trust a third party in such case?



One solution that have been found by the eclair android wallet is to forbid relaying or receiving payments (i.e. you can only send payments). That way an outdated transaction is always in defavor of the fraudulent node. This is obviously not satisfying if you want to have the full functionalities of the Lightning Network.
(disclaimer: I'm not an expert on Lightning)

I believe the design for Watchtowers (the services which watch for other parties trying to push an old state to the chain) is such that they're not actually given the channel close transaction.

Instead, they're giving half of the txid to watch for, and an encrypted version of the channel close, using the full txid of the watched transaction as key.

This means they are not actually able to dishonestly close your channel. They can only do so when an invalid state update is published.