The endomorphism-algebra and tangle-invariant conjecture for Khovanov homology

For each natural number $m$, let $T_{2m}$ be the full projective tilting module in the relevant parabolic category ${\mathcal O}$ for $\mathfrak{sl}_{2m}$, and let ${\mathcal H}_m$ denote Khovanov's algebra. For a $(2m,2n)$-tangle $t$, let $\Phi^{or}(t)$ be the associated functor and let $\check{X}(\Phi^{or}(t))$ be the corresponding complex of bimodules. Endomorphism-algebra and tangle-invariant conjecture. For any natural number $m$, there is an isomorphism of algebras

$$p_m:\operatorname{End}_{\mathfrak g}(T_{2m})\cong {\mathcal H}_m.$$

Moreover, the homological tangle invariant

$$t\mapsto {\bf H}^{\bullet}(\check{X}_{\Phi^{or}(t)})$$

is Khovanov's invariant. This conjecture would identify the tilting-module realization of the parabolic representation-theoretic construction with Khovanov's algebra and invariant. The supplied text gives no resolution status.