None of them require auth (42). I just checked my personal relay.tools relay which is one of my configured inbox relays, and it's got no entries for kind 9734 or 9735.
Additionally, I tried adding relay.coinos.io to my public inbox config, and I'm getting the same result: the purple wheel gets to about 95% complete and it just hangs. Checking my coinos.io wallet, there's no corresponding zap transaction. And checking my personal relay records show no corresponding 9734 or 9735 entries.
Do you know if any other coinos.io users have successfully zapped via NWC in Amethyst? I think it would be helpful to know whether this is a systemic issue, or some weird thing specific to me.