well I have not checked but I have a very early savage, I'll have to dig it out.
the later 6 groove barrels were manufactured the same as any other Enfield barrel, with the exception being that they were rifled on a machine designed to rifle BREN barrels.
As you say the story that "Bren barrels" were used on no4s is BS.
BREN barrels are substantially shorter in length than no4 barrels.
Savage manufactured 6 groove left hand barrels in 1941 and early 1942.
Long Branch manufactured 6 groove right hand barrels from 1949 thru 1958.
I have samples of both of these in my collection.
So to recap:
2 groove (MkII barrel) very common war time barrel
3 groove (rare trials) documented but none observed to my knowledge
4 groove (authorized to Savage & found on "dispersal" no1 rifles) I have owned both (also some post war "malasian replacement barrels are 4 groove)
5 groove (standard mk1 barrel) & 2 piece "dolled"(early hammer forged) 2 piece barrels are documented but none observed yet.
6 groove (left hand, early '41/'42 Savage)(right hand C.Mk4 CAL '49-'58) i have samples of both these barrels.
One reason that the war time 3, 4 & 6 (and 2 piece) groove barrels are hard to find, in 1945 thru 1956, every British No4 rifle was FTR'd and the "non standard" barrels were replaced with standard 5 groove barrels or the guns were destroyed.