I found a Japanese site that says that the later versions of the Type Hei, including the long and short versions, were designed for the 7.7×58mm Arisaki cartridge.
http://www.horae.dti.ne.jp/~fuwe1a/newpage71.html
Also I will leave as you say a Russian source, actually Zwershitz Andrei is a specialist in Japanese small arms, so it is a good source.
Although I can guess that there were two versions of the later Type Hei, both the 6.5mm version and the 7.7mm version.
Thanks. It is not this website written in russian but this one looks interesting too. I don’t really remember where the website I think of was, but I remember that all the pictures in the article written in russian had forgottenweapons as the source for the pics. I will try to find it back when I have time, I should be able to find the topic somewhere in my posts history. 
I discounted a clipping of the article, here is the full article :
Yes, that’s the one. You’ve been quick. 
To be honest, I think I got it wrong and was talking about Type 4, not the later Type Hei. On the other hand, I haven’t heard about the short and long version of Type 4. Anyway, it’s very complicated as always.
