I have to make a correction to my earlier post. I incorrectly thought the Jedi helmet used a different chin strap. But I over-looked a photo which clearly shows the same chin strap/cup. In fact, it looks like they used the attachment pulled straight out the helmet, even with the leather strip at the end.
The ESB attachment has been modified a bit, as they moved the d-rings to the very end of the strip that comes out of the slit.
It came from a British SWAT-type helmet.