Borie's bijection conjecture for 1234-avoiding up-down permutations

Let $n$ be a positive integer. An up-down permutation of size $2n$ is a permutation of $\{1,\ldots,2n\}$ whose successive entries alternately rise and fall, and a permutation avoids $1234$ if it has no increasing subsequence of length four. Let a weighted Dyck path be a Dyck path together with the weights described in Borie's construction; denote the relevant class by $WD_{2n}$. Borie's conjecture. There exists a combinatorial bijection between up-down permutations of size $2n$ avoiding $1234$ and a certain class of weighted Dyck paths. This conjecture proposes a new correspondence between two families counted by the three-dimensional Catalan numbers. The paper presents a more structured bijection resolving this existence claim, so the original conjecture is solved.