The idempotent truncation conjecture for blob algebras and Soergel endomorphism algebras

Let $\Lambda_n$ be the indexing set of weights, let $W$ be the relevant Weyl group, and let $\mathcal{A}^w$ denote the orbit or alcove associated with $w\in W$. For $\lambda\in\Lambda_n$, let $b_n(\lambda)=e(\boldsymbol{i}^\lambda)b_n e(\boldsymbol{i}^\lambda)$ be the idempotent truncation of the blob algebra $b_n$, and let $A_w$ be the associated graded diagrammatic algebra. The idempotent truncation conjecture. If $\lambda\in\mathcal{A}^w$, then

$$b_n(\lambda)\cong A_w$$

as graded cellular algebras. The conjecture proposes that these two graded cellular algebras provide the same algebraic model for the corresponding weight and Weyl-group element, explaining the equality between their graded decomposition numbers and the relevant Kazhdan–Lusztig polynomials. Its status is not resolved in the supplied source.