Make three useful lanes
Separate conversations someone can answer now, work awaiting an internal decision and cases waiting for the customer. Write a short definition for each lane so two agents sort the same example consistently. Choose a visible owner even for cases that are waiting.
Add a due time when there is a genuine commitment, such as a promised callback. Avoid turning every message into an emergency; that hides the promises that really do expire.
Read the latest message in context
Before assigning a case, check the last promise, the customer’s current question and any action already taken. A short “yes” may confirm an appointment or approve an unrelated change. The transcript needs enough context to tell the difference.
Ask each candidate inbox to demonstrate assignment, reassignment and searching for an older exchange. Record the steps an agent has to perform, rather than judging the demo only by the presence of an assignment icon.
Review the work that stays behind
At the end of a shift, inspect unowned cases and internal decisions that have stalled. Give the next shift a concise note with the unresolved action and the person who can approve it.
For your pilot, count reopened cases and missed promises alongside response time. Fast replies are helpful when they also move the work forward. Keep this scorecard small enough that the team will maintain it.