This is now proven both as Theorem 1.7 in my paper "[Knot invariants and higher representation theory I](http://front.math.ucdavis.edu/1001.2020)" and by Kang and Kashiwara in "[Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras](http://front.math.ucdavis.edu/1102.4677)."