This idea that there is inferior fuel floating about on the commodities market is nonsensical.
My BIL is an oil broker and says that everyone gets their fuel from the same refineries. The brands have their own storage and, sometimes, processing but they will buy from another source,if it's cheaper than theirs,too.
I can't see ASDA knowingly selling suspect diesel or buying it from some hooky operator.
I'm not sure about supermarkets but I have seen BP road tankers drawing fuel from the Texaco depot in Fishersgate which makes me think that fuel may be collected from a local rather than a branded depot. A bit like regardless of who you buy gas from it all comes through the same main.