On the permutations that strongly avoid the pattern 312 or 231

In 2019, B\'ona and Smith introduced the notion of strong pattern avoidance, that is, a permutation and its square both avoid a given pattern. In this paper, we enumerate the set of permutations π which not only strongly avoid the pattern $312$ or $231$ but also avoid the pattern τ, for τ∈ S₃ and some τ∈ S₄. One of them is to give a positive answer to a conjecture of Archer and Geary.