One of the largest UX issues with non-custodial mobile lightning wallets is being limited to only receiving payments while having the wallet open. This pushes users to custodial wallets.

Lightning mobile push payments is a plan to use a push notification to wake up mobile apps to come online to receive a payment.

So the payment is still fully P2P and no 3rd parties need to hold any user funds. Each wallet provider will just need to run their own push notification server.