The orientifold LLT polynomial formula for graded decomposition numbers

Let $\Lambda_n$ index the cell modules $\Delta(\lambda)$ and simple modules $L(\mu)$ of the orientifold quiver Temperley--Lieb algebra, and let $\langle k\rangle$ denote grading shift. For $\lambda,\mu\in\Lambda_n$, let $n_{\lambda,\mu}(v)$ be the orientifold LLT polynomial obtained from the stated factorisation of the tableaux-counting matrix. Orientifold LLT polynomial conjecture. Over the complex field, the graded decomposition numbers are given by

$$\sum_{k\in\mathbb Z}[\Delta(\lambda):L(\mu)\langle k\rangle]v^k=n_{\lambda,\mu}(v).$$

This gives a tableaux-theoretic description of the graded decomposition matrix of the orientifold quiver Temperley--Lieb algebra. The source records that the corresponding decomposition matrices for generalised blob algebras were conjectured previously and later proved; the supplied status evidence marks this statement as resolved.