On the set partitions that require maximum sorts through the $aba-$avoiding stack

Recently, Xia introduced a deterministic variation φσ of Defant and Kravitz's stack-sorting maps for set partitions and showed that any set partition p is sorted by φN(p)aba, where $N(p)$ is the number of distinct alphabets in p. Xia then asked which set partitions p are not sorted by φabaN(p)-1. In this note, we prove that the minimal length of a set partition p that is not sorted by φabaN(p)-1 is $2N(p)$. Then we show that there is only one set partition of length $2N(p)$ and N(p) + 1 2 + 2N(p) 2 set partitions of length $2N(p)+1$ that are not sorted by φabaN(p)-1.