Geometric categorification conjecture for tensor products of \mathfrak{sl}_2-modules

Let $W$ be the Weyl group, let $S_{\bf d}$ be the subgroup associated with the tensor-product datum ${\bf d}$, and let $H_{\bf d}^i=(B^{\bf d}\otimes C)^{W_i}$ and $H_{\bf d}^{i,i+1}=(B^{\bf d}\otimes C)^{W_{i,i+1}}$. Set

$$\mathcal{C}_{geom}:=\bigoplus_{i=0}^{n} H_{\bf d}^i\operatorname{-mod}.$$

Let $E_{geom}$ and $F_{geom}$ be the direct sums of the induction functors $E_i$ and $F_i$ between these module categories, and let $\Phi_1,\Phi_2:{\bf G}(\mathcal{C}_{geom})\to\overline V_{\bf d}$ be the vector-space isomorphisms sending the simple and indecomposable projective bases, respectively, to the corresponding bases of $\overline V_{\bf d}$. Geometric categorification conjecture. The isomorphisms $\Phi_1$ and $\Phi_2$ agree and are isomorphisms of $\mathfrak{sl}_2$-modules, where the action on the left-hand side is induced by the functors $E_{geom}$ and $F_{geom}$. The conjecture asserts that this geometric construction categorifies the tensor-product representation; the supplied text gives no resolution status.