A snoozed conversation now leaves the Inbox list until its time comes — or until the customer writes again, which wakes it at once. While it sleeps it is left out of Needs you now, the unread numbers and the Overview. Reply-time targets still count.
A search still finds a snoozed conversation, and a view with Snoozed set to Yes lists them all. In views, No now filters too.
Search conversations… keeps every result: a word inside any message, or a conversation number such as #4611.
The phone app reads the same list.