Default on report: my name was used fraudulently to buy a mobile phone and contract

Doveinflames25

Hi, 
I was in the process of applying for an AIP and decided to look at my credit report. This is not something I had been savvy before and it was my first time checking.
I then see there are a number of defaults by Virgin, as someone purchased a phone and set up a contract under my name.

Back in 2019 I received a letter to my name and address welcoming me to Virgin with the details of the purchase. The bank account shown under my name didn't belong to me. I had not bought a phone, nor taken a contract with Virgin. I contacted them and they told me they will solve it. I also contacted Monzo who reassured me it was not under my name. Naive me I thought that was the end of it.
A couple of months later I moved house.
Some time after that I get a bunch of letters from my old address, one of them being from a debt collection agency, about this Virgin contract. I called Virgin again and they reassure me it is all dealt with. No more debt collection letters after that.

Now, March 2023 I check my credit report and see there are 5 defaults for Virgin phone contracts. I've been in touch with Virgin (paper trail this time), and issued them all the relevant information I have. They confirmed they had in record that I flagged this in 2019, but for some reason it wasn't closed. They apologise but all I want it's for it to be solved.
i'm finding the process very stresful and convoluted. I contacted them three times now to follow up as I don't hear back in the timeframe they tell me, and every time they give me a different timeline.

This is obviously witholding the process of purchasing our family home, and it is truly stresful. 
I'm fully aware I should have been on top of my finances back in the day, but one can only learn (the hard way this time!).

I would love to hear what I should do next. Would engaging a solicitor help? What type of information and to whom could I send to move this forward in a positive way?
Any advice would be really appreciated. 
Thanks

MorningcoffeeIV

You don't need a solicitor.

Escalate a formal complaint if not already done so, then FOS if not resolved.