Actually we can know. I am hoping to find out in for my thesis
Some of the older LHW from our club were actually bought from Walter, none have tested positive. Plus we have all of Walters records (take that at face value) Regular whippets DO carry a long haired gene. So do many other breeds. Its polygenetic and recesive so can hide for a long time in a large population. Everyone agrees that whippets used to come in three coat types. The wire hair was lost because its a dominant trait and can't hide in the genome. Shelties have a double coat along with the length, LHW never show up with a double coat, putting the idea of sheltie as a contributor as an unlikely source.
Personally I don't care if something was mixed in. BUt I find it amusing that so many people are adamant that Walter HAD to mix things in. Genetically its not at all unreasonable to do what he did and NOT mix things in.