What about DM inbox and outbox relays?
There's not much of free relays for DMs . Not all clients support specific DM relay setting. Anyway I believe most of us using free DM relays like auth.nostr1.com. 0xChat have it's own relay.0xchat.com (you can add it to your client easily). From my 0xChat app: "If not set, [DMs] will be sent to the general relay by default".