Greedy bubble factorization conjecture for permutation matrices

Let $P$ be a finite permutation matrix of bandwidth $w>0$. A greedy bubble matrix of $P$ is a product of reducing and overtaking swaps of $P$ such that $P$ and the resulting matrix $GP$ have no common inverted pairs. Greedy bubble factorization conjecture. $P$ is the product of fewer than $2w$ greedy bubble matrices. The conjecture proposes a bound on the number of greedy bubble steps needed to factor a permutation matrix, building on computations for permutation matrices of order at most $9$; the supplied source gives no resolution status.