A family of bijections preserving generalized equivalence classes

Let $w$ and $u$ be permutations, and let $\theta^{(k)}$ denote a bijection on permutations for each $k$. The bijections are required to satisfy the properties stated in Propositions and, and $\sim_k$ denotes $k$-equivalence. Bijection-family conjecture. There exists a family of bijections $\theta^{(k)}$ on permutations satisfying Propositions

$$and$$

such that, whenever $w \sim_{k-1} u$, one has

$$\theta^{(k)}(w) \sim_k \theta^{(k)}(u).$$

Such a family would express each $k$-equivalence class through images of $(k-1)$-equivalence classes and, as indicated in the source, would yield a simple proof of Macdonald positivity.