This is now proven both as Theorem 1.7 in my paper "Knot invariants and higher representation theory I" and by Kang and Kashiwara in "Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras."

This is now proven both as Theorem 1.7 in my paper "Knot invariants and higher representation theory I" and by Kang and Kashiwara in "Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras."

This actually seems to follow from Theorem 5.16 of Rouquier's paper on 2-Kac-Moody algebras.

This is now proven both as Theorem 1.7 in my paper "Knot invariants and higher representation theory I" and by Kang and Kashiwara in "Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras."