Decidability of parity and Wilf-equivalence for single forbidden permutations

Let $\mathcal{F}$ and $\mathcal{F}'$ be forbidden sets of permutation patterns with $|\mathcal{F}|=|\mathcal{F}'|=1$. Single-permutation decidability conjecture. Both the parity problem for $C_n(\mathcal{F})$ and the Wilf-equivalence problem for $C_n(\mathcal{F})$ and $C_n(\mathcal{F}')$ are decidable. The paper presents this as a belief about a restricted case of the preceding open problems, in contrast with the conjectured undecidability for arbitrary finite forbidden sets.