No, this story take place after. In the other manga Kaoru is still wearing multiple rings. After Hara gave him a ring, Kaoru stopped wearing all his other rings.
Oh right!! He was still wearing multiple rings....
Somewhat kind of spoilers....
but maybe the author forgot about that one... since this story is made after the first one and not the other way around... as i have heard, Authors are bad at remembering every single detail they put into a story... The impact of Akai is what proves the point of Hara... You can cut the string but if your feelings are involved and the string reconnects then that could be forgotten.... And Kaoru is too laid back to give something to the uke of Akai the scissors, which cuts the string.... only thing that he never dares to cut...
That is just my analysis though
..... pages 17 and onwards proves that this story plot happened BEFORE the original series... Akai Ito No shikkou yuuyo... You cannot give an advice and explanation to someone when you yourself still fears about string of fates and how powerful it connects people..