The slN-web algebras and dual canonical bases

Using this Morita equivalence and Brundan and Kleshchev's [4] work on cyclotomic KLR-algebras, I show that there exists an isomorphism between a certain space of slN-webs and the split Grothendieck group of the corresponding slN-web algebra, which maps the dual canonical basis elements to the Grothendieck classes of the indecomposable projective modules (with a certain normalization of their grading).