The task has been completed, and it does not go away. Where to look in the settings?
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I suspect that the issue has no resolution set. An empty resolution tells Jira an issue is unresolved and needs more attention, but any value in it means resolved.
Have a look at the plain issue view, I suspect you'll find the resolution says unresolved - that's what Jira displays for an empty resolution.
This happens when your workflow does not account for the resolution properly. A transition into any of the last (end) status should set a resolution either by asking the person closing it, or by a post-function.
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