claiming someone is close minded is a double logical fallacy, IMO. Assuming that you haven’t said something like, I am not interested in any new information on that topic.
First, they are asking for special pleading. They are saying their thing is not subject to the regular rules of evidence and reason. Second it is an ad hominem. There is something wrong with you, the listener, you are unable to process information correctly, there is something wrong with your mind.
Open mindedness is a quality that exhibits willingness to consider a point of view. It does not require responding to the same evidence repeatedly or repeating something that has been well published. It can take practice to catch the difference between open minded and gullible.