Wide-equivalence conjecture for the branching functor

Let $R^athfrak{g}_n]^ambda]$-mod be the category of modules over the cyclotomic KLR algebra indexed by the dominant weight $\lambda$, and let

$$\Pi^{\lambda}=\operatorname{ext}^{\lambda}$$

be the branching functor defined by extension of scalars to the direct sum of the algebras $R^{\xi_{(\mu,\nu)}(\lambda)}_{\mathfrak{g}_{n-1}}$. A functor is a wide equivalence of categories when it is full and a bijection on objects.

Wide-equivalence conjecture. The functor $\Pi^{\lambda}$ is injective on objects and therefore a wide equivalence of categories.

The preceding result establishes that $\Pi^{\lambda}$ is full, essentially surjective and intertwines the categorical $\mathfrak{g}_{n-1}$-action, while the conjecture asks for injectivity on objects as well. The parser supplies no evidence that this conjecture has been resolved.