Breez does routing on the phone so no issue there (except unencrypted backup to Google by default). But of course if your phone hasn't finished downloading the graph, your routing might misbehave and your payments might fizzle. There are tradeoffs everywhere. 

energycurrency

bitcoin

deleted by author

> it's self custodial?

Yes in the sense that you do hold your keys and your end of an actual channel. However:
 - ACINQ does the routing, so you have no privacy and they can in principle refuse to process your payment
 - Like with any other Lightning channel there's a theoretical possibility of a fraudulent settlement that you need a watchtower to defend from. IMHO that is not a concern because if ACINQ does that and you manage to open the wallet at the right time, you'll expose them to the whole world.