Answer
To be able to edit or quick-edit a task you must either:
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Be an admin (i.e., have the "key")
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Have the permission while the task is open
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Have the permission while the task is not approved, and you are a team leader
In the 3rd example, it is important to ensure that:
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The user who wants to edit the task is the team leader for the team the task is assigned to.
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Alternatively, change the team on the task.