Two refined Wilf-equivalences for patterns of length 4 and a conjecture on Dumont permutations of the first kind

We construct a bijection between $2143$-avoiding and $3421$-avoiding permutations that preserves the descent top set and the ascent top set and reverses the left-to-right order of the peaks. This bijection allows us to establish a bijection between $1243$-avoiding and $3412$-avoiding permutations with the same properties. These results refine two cases of a conjecture posed by Burstein and prove a conjecture posed by Burstein and Jones.