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encemil I spoke to a developer about this and he said that the timer should be triggered only after the agent has sent their response. the timer's main function is to clear conversations from view that have already been acted upon so that agents address their assigned conversations in a timely manner. agents should not be pushed another conversation while in the midst of drafting a response.
furthermore, if the agent is at capacity after sending their response, the next conversation the timer pushes to them should be a conversation that already exists in their Assigned to Me queue and not a new conversation from the Available queue. in other words, the timer should be adhering to the Stop pushing from Available when agent has ___ conversations assigned to them setting.
it sounds like this is not the behavior you're seeing in your instance, so what you're describing may very well be a bug. the developer I spoke to also mentioned that there's an open Jira ticket to address some erroneous behavior with how the timer is being triggered, so your issue could be related.
I recommend submitting a ticket to Support with this information so that they can assist you further. thank you for bringing this to our attention!