The Steep-Bounce bijection conjecture

Fix integers $n$ and $k$ with $k\le n$. A Dyck path is a lattice path of semilength $n$ from $(0,0)$ to $(n,n)$ that stays weakly on one side of its diagonal; a nested pair $(\pi_1,\pi_2)$ consists of Dyck paths of size $n$ with the prescribed nesting relation. A bounce path has $k$ parts, and a steep path has isolated east steps strictly below the top. Steep-Bounce conjecture. For any $k\le n$, there is a bijection between nested pairs $(\pi_1,\pi_2)$ of Dyck paths of size $n$ such that $\pi_1$ is a bounce path with $k$ parts, and nested pairs $(\pi'_1,\pi'_2)$ of Dyck paths of size $n$ such that $\pi'_2$ is a steep path with $n-k$ isolated east steps strictly below the top. This reformulates the refined pipe-dream and walk conjecture using the stated bijections with colored Dyck paths and nested pairs; the proposed bijection remains open in the source.