Grothendieck-group realization of irreducible representations by categorical modules

Let $\lambda\rightarrow\Lambda^+(n,d)$ be dominant, let $\mathcal V_{\lambda}$ be the categorical module associated with $\lambda$, and let $dot{\mathcal V}_{\lambda}$ be its Karoubi envelope. Let $V_{\lambda}$ denote the corresponding irreducible representation. Irreducible-module Grothendieck-group conjecture.

$$K_0^{\mathbb{Q}(q)}(\dot{\mathcal V}_{\lambda})\cong V_{\lambda}.$$

The source has already proved surjectivity of the natural map from $V_{\lambda}$ to this Grothendieck group, so the conjecture amounts to the missing injectivity and would realize the irreducible representation categorically. The source gives no resolution.