The refined pipe-dream and quarter-plane walk bijection conjecture

Fix integers $n$ and $k$. Consider pipe dreams $P$ with $\omega_P\in\mathfrak{S}_n$ whose factorization into atomics consists of $k$ identity permutations, and quarter-plane walks starting at $(0,0)$, ending on the horizontal axis, consisting of $2n$ steps from $\{(-1,1),(1,-1),(0,1)\}$, with exactly $k$ steps equal to $(0,1)$. Refined pipe-dream and walk conjecture. These two families have the same cardinality. This is a stronger refinement of the preceding dimension conjecture and is stated as a conjectural bijective correspondence between the two families.