On the permutations that strongly avoid the pattern 312 or 231

In 2019, Bóna and Smith introduced the notion of \emph{strong pattern avoidance}, that is, a permutation and its square both avoid a given pattern. In this paper, we enumerate the set of permutations $\pi$ which not only strongly avoid the pattern $312$ or $231$ but also avoid the pattern $\tau$, for $\tau \in S_3$ and some $\tau \in S_4$. One of them is to give a positive answer to a conjecture of Archer and Geary.