Close but no cigar! The one on ebay is a Champion cap and I see them often. As Rik stated in his previous thread, the KLG caps must have come with two different thread inserts. Here is a KLG cap with the correct period KLG L777 plug as fitted to BSA's which has a 3BA thread. Next to it is plug with a bigger 2BA thread. Modern plugs of course have a 4mm thread. Unfortunately both the KLG caps I have, have the 2BA inserts which makes them fit for display only. Ron
KLG caps came with three different thread types, Ron - 0 BA as well.
This page from the 1933 catalogue gives a good overview. Unfortunately, the L.777 was not yet in the range. Most motorcycle and British car plugs seem to have been 3BA and this was pretty well standardised by 1937 or so which is the first record that I have of the L777.