Graded cellularity and the monotonicity conjecture

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 interpretation, we can provide a proof of the monotonicity conjecture for any Coxeter system.