First, whatever you may think, DO NOT roll over and give in to the dispute.

If you do that, it's a guaranteed refund.

So you challenge it.  After all, the worst that can happen is that you end up in same situation. ie refund.

You already have an email from the customer stating that it is now working.
You can attach this to the dispute.  If I remember correctly, you can add files in your defence.

State the procedure that you have followed, when sending the item, along with POD.

Give a brief summary of what has happened and what you have done to fix it.
Say that the buyer has not followed proper procedure, in opening a return in order to give a refund.
And if you can think of anything else that is relevant, add it to the case.

And yes of course the buyer can close a dispute.  He just contact his bank and tells them all is ok now.