Digital interactions handled from "All messages" inbox no longer remain in triage queue.
For many companies, multiple agents work across multiple channels often using "all messages" tab to complete queries inbetween tasks rather than setting as available.
Currently, even when a query has had a response through "all messages" it remains in the queue to be accepted through "my messages". If an agent assigns a conversation to themselves through "all messages" then responds, this should remove it from the queue.
It isn't productive to need to accept conversations through "my messages" which have already been responded to by another member of the team.

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Rebecca commented
The primary reason we must prevent conversations that have already received a response from routing to agents is that our leadership monitors the queue to ensure we meet our contracted SLA. However, this can sometimes be counterproductive. When there are many messages in the queue that leadership resolve, agents are then required to accept these through routing mode to properly close and resolve them.
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This idea will be taken under consideration, thanks for this interesting feedback.
A workaround exists already to exclude a certain portion of the backlog from "My messages" is to create queues that will target only specific categories/channels/languages. Meaning that a certain portion of messages, will not be routed to agents and be only addressable in the "All messages" mode.
Would this work or do you have a specific reason to mutualise the processing of the same backlog subset in both modes (cherry-picking and routing)