  The graded cellularity of Libedinsky Double Leaves, which form a basis for
the endomorphism ring of the Bott_Samelson_Soergel bimodules, allows us to view
the Kazhdan_Lusztig polynomials as graded decomposition numbers. Using this
point of view, I provide in this paper a new proof of the monotonicity
conjecture for the Kazhdan_Lusztig polynomials of any Coxeter system.
