Egge triples and unbalanced Wilf-equivalence

Egge conjectured that permutations avoiding the set of patterns $\{2143,3142,\tau\}$, where $\tau\in\{246135,254613,263514,524361,546132\}$, are enumerated by the large Schr\"oder numbers. Consequently, $\{2143,3142,\tau\}$ with $\tau$ as above is Wilf-equivalent to the set of patterns $\{2413,3142\}$. Burstein and Pantone proved the case of $\tau=246135$. We prove the remaining four cases. As a byproduct of our proof, we also enumerate the case $\tau=4132$.