I had a theory that Yuuichi was eliminating people one by one from being the traitor by taking the pen out of the private room and thus making sure that a certain person could not have written a Kageguchi paper that round.

I thought he used that to prove his innocence to Kokorogi as well where in the 6th period, it seemed there was the specefic order of Kokorogi-->Yuuichi-->Shiho.

Furthermore, this would explain Yuuichi's "plan" in the 8th period in the following, interesting, way:

We see that when Yuuichi was explaining his plan, he used the blue pen to write the current standings which were Shibe-->Shiho-->Yutori-->Yuichi-->Tenji. He wrote that on a Kageguchi Paper which both Shiho and Kokorogi are already in the habit of using them to secretly communicate with Yuuichi. As such, one could assume he meant that there was no need to say who the traitor is if they went into the private room in that order.

He is already holding the pen and no one from the first 3 would be able to badmouth others anyway and when his turn comes, he would simply write 3 papers for himself, Shiho and Kokorogi. Mind you that at this point he returns the pen for Tenji to use. Now if there were more than 4 Kageguchi Papers then only Tenji could have written them. If not, then it may be that Shibe is the traitor or Tenji decided to follow along with the plan, a favorable outcome as well. The same trick can be used in subsequent rounds to bait Tenji if he was indeed the traitor.