Character map conjecture for tilting modules in \mathcal O_{m|n}

Let $\mathcal O_{m|n}^\Delta$ be the subcategory of objects in $\mathcal O_{m|n}$ admitting Verma flags, let $K(\mathcal O_{m|n}^\Delta)$ be its Grothendieck group, and let $\mathscr T^{m|n}_{\mathbb Z}$ be the target based module. For $\lambda\in X(m|n)$, write $T(\lambda)$ for the corresponding tilting module and $f_\lambda$ for its associated index.

Character map conjecture. If $i:K(\mathcal O_{m|n}^\Delta)\to\mathscr T^{m|n}_{\mathbb Z}$ is the map defined in the cited theorem, then

$$i([T(\lambda)])=T_{f_\lambda}(1)$$

for every $\lambda\in X(m|n)$.

This conjecture is motivated by the theorem preceding its formulation and predicts that the image of each tilting-module class under the character map is obtained by specializing the corresponding canonical-basis element at $q=1$. Its resolution depends on identifying the representation-theoretic map with the combinatorial basis construction.