Hello gentlemen, I've talked to Kanoyama about this. Apparently, the traditional use of the linen in Japan is just to remove oils from the blade. So the linen gets much less use than the leather, apparently--thus, the linen, being separate from the other strops, just gets grommet. He seemed a bit consternated that I would ask him to add something to the linen at all. He said "Just put it on a hook."

Sooo....I guess just put it on a hook?

I put a strap in mine and hung it on the same hook as the leather, and it works pretty good.