They would, and all the evidence of the violation. I'm going to write to the DVLA and ask if they could contact the owner on my behalf with a request that they contact me about the Ducati.
Hi Tigler, please keep us up to date with how you get on. I had to sell a bike about 12 years ago just for the cash, as I was out of work at the time, and would love to track the owner down and buy it back.
Apparently it's unlicensed at present, not even on SORN. Makes me wonder if it's gone to the great scrapyard in the sky; hope not.