Compositional equivalences for pairs of length-four patterns

Let $C_n(\tau,\tau')$ be the set of all compositions of $n$ that avoid the patterns $\tau$ and $\tau'$, and let $c_n(\tau,\tau')=|C_n(\tau,\tau')|$. Compositional equivalence conjecture. If $n\geq 1$, then

$$c_n(121,1342)=c_n(121,3142) \quad\text{and}\quad c_n(212,3124)=c_n(212,3142).$$

These two equalities assert compositional equivalences that the authors' enumeration data suggest but that they are unable to prove by their present methods.