New comments should reopen tickets
If you enable the "reopen ticket" rule, the submiter is no longer able to close tickets, so this is not a solution.
new comments (like replying back to the "close case" template) should cause the ticket to reopen

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Just so I am clear, let me know if this scenario captures your request. There is a ticket rule that can be enabled that will automatically reopen a ticket should the enduser reply to a closed ticket email notification with something like “Hey don’t close this out, I’m still not fixed.” The ticket rule engages and then automatically changes the ticket to “reopened”.
But let’s say that enduser said “thank you, great job” – technically we will still reopen the ticket since the rule is doing exactly what is expected. The customer wants the enduser to be able to close it out at that point and we will not allow the enduser to do so. Only the ticket owner can close it out. That is working as designed.
If this is the case, the best option would be to just turn that rule off and not use it. You can make the Status field an “editable” field and the enduser can close this ticket out from the portal if they need to or reopen if they need to. Will that help - even temporarily? I ask because there are so many higher rated requests, that I am not confident that we can move this into a relase anytime soon.