Parity problem for pattern-avoidance sequences

For a finite set of permutation patterns $\mathcal{F}$, let $C_n(\mathcal{F})$ denote the number of permutations in $S_n$ avoiding all patterns in $\mathcal{F}$. Parity problem. The problem of deciding whether

$$C_n(\mathcal{F})\equiv 0\pmod 2$$

for every $n\in\mathbb N$ is undecidable. The authors present this as an open problem extending their undecidability theorem and expect their methods may eventually prove it, although substantially more work is required.