Strang's bandwidth factorization conjecture for permutation matrices

Let $P$ be a finite permutation matrix of bandwidth $w>0$, where bandwidth $1$ means tridiagonal. A bandwidth--1 permutation matrix has nonzero entries only on the main diagonal and the two adjacent diagonals. Strang's conjecture. $P$ can be written as the product of at most $2w-1$ bandwidth--1 permutation matrices. This conjecture asks for a uniform sharp factorization bound in terms of bandwidth; the supplied source gives no resolution status.