Advice on very old potential debt

In late 2014, I received correspondence from a DCA acting on behalf of Barclays, pursuing a student overdraft of around £1k from 2003. I hadn't used this account in many years, well over six, and it is my belief (though, in full honesty, I can't actually remember, and can't prove anything) that it was closed a couple of years later in around 2004/2005/2006 (and it wasn't on my credit file in 2014), and the overdraft transferred to RBS (an account also closed so long ago that it is long since gone from my credit file. I certainly don't recall using it after 2004, and it was quite common then to transfer student accounts/overdrafts between banks. I emailed the DCA, stating that I didn't acknowledge the debt, that it would be statute-barred in any case, and requested an original credit agreement. None was forthcoming, and after some back and forth, I invited them to issue proceedings if they felt they had an enforceable credit agreement. They did not respond. I am currently looking at applying for a mortgage, and have noticed that whilst this account doesn't appear of Equifax or Experian, it is on my TransUnion file, showing as set up in 2003 and defaulted in 2015. Does anyone have any advice as to what I should do? The 2019 Doyle case might make my contention that it was statute-barred somewhat shakier, though, equally, I haven't heard anything about it in over five years, I cannot believe that, if the account was not closed/settled in the mid-2000s as I believe, that Barclays would not have issued a default notice before 2015. Does anyone have any advice as to what I should do? I want this off my credit file, even if the account wasn't closed in 2004, I believe that Barclays/its DCA are trying to pull a fast one by recording a default in January 2015, but given that neither party has any records with which to validate their belief/position, it hardly seems fair that I pay out for something that either doesn't exist or ought to be statute barred?

A further point is that whilst it is Barclays, the account was opened at a branch in Scotland (where I lived/studied), where I believe that the Limitation period is five years, so, even if it transpired that the default was legitimately in 2015, would the matter now be statute barred anyway? Or would it be six years given that I now live in England?