Kisaragi Gentaro's uniform - red chain?

I bought the official top that Kisaragi Gentaro wears and upon inspection, I don't see the red chain thing behind the buttons where Gentaro has on his. This is what I'm taking about;



See the red chain behind the buttons? Is that something Gentaro the character of the show put on his uniform or do all uniforms worn in Asia suppose to have something like that? I've seen some japanese cosplayers have it on theirs.

I can supply some more screencaps, from the KR/SS anniversary talk and show:

The other side of the jacket (there's clearly nothing on that side) -
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Images where you can somewhat see the chain/cord/whatever it is -
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You can't see it clearly, but it does shine when it catches the light, so it's not a matte color. It appears to start at the collar, and loops at the buttons. It's pretty loose, since it bounces around and can be seen from the outside of the jacket.