I too have had the gun show dealer who is an "expert" in all things 40X try to convince me that the "all original 40X's" he had were equipped with "factory" Hart barrels, and went on to show me the heart shaped proof mark as well. As we all know, Remington used that proof mark on many barrels produced by Remington itself, and this in no way indicates the barrel in question to be one produced by Hart.
I doubt that any substantial number of 40X's were ever equipped with Hart barrels from the factory, though I would believe that in special instances, and those are likely rare, that Remington may have substituted an aftermarket barrel, and possibly a Hart, on some of it's custom builds that required barrels with specs that they did not normally produce. But as far as there being any large number of 40X's with Hart barrels, I seriously doubt it. Though on another note, I have also been told by some older fellows I know, that back in the day, Remington would build you a custom rifle with basically any parts you wanted, and I would think if you were willing to spend the money, they would likely supply you with a barrel from whatever manufacturer you desired.
But, if this is true that Remington did purchase a number of barrels from Hart, then I would think that Hart would have invoice records of these transactions, or at least have some recollection of selling a number of barrels to Remington.
I feel that if there were a large number of Hart equipped 40X's produced, there would be more record of it somewhere. If there are any of the fabled Hart 40X's in existence, I think they are few and far between, with the number produced likely being less than 1000 of all those ever produced. If there were more, there would be some type of record in existence to prove otherwise, at least one would think so anyway.
This is all speculation and I have no real proof, but I just don't believe there to be "thousands" of Hart barreled 40X's.
If there are, I hope some proof can materialize and put this myth or legend to rest.