The decategorified blob versus Kazhdan–Lusztig conjecture

Let $P_1^l(n)$ be the set of $l$-tuples of non-negative integers summing to $n$. Let $\boldsymbol{\lambda},\boldsymbol{\mu}\in P_1^l(n)$ be regular, with $\boldsymbol{\mu}$ in the orbit of $\boldsymbol{\lambda}$, and let $w_{\boldsymbol{\lambda}}$ and $w_{\boldsymbol{\mu}}$ be the associated affine Weyl-group elements. Write $d^p_{\boldsymbol{\lambda},\boldsymbol{\mu}}(v)$ for the graded decomposition number and $h^p_{w_{\boldsymbol{\lambda}},w_{\boldsymbol{\mu}}}(v)$ for the corresponding $p$-Kazhdan–Lusztig polynomial. Decategorified blob versus Kazhdan–Lusztig conjecture.

$$h^p_{w_{\boldsymbol{\lambda}},w_{\boldsymbol{\mu}}}(v)=d^p_{\boldsymbol{\lambda},\boldsymbol{\mu}}(v)\in\mathbb{Z}[v,v^{-1}].$$

This is the decategorified form of the main categorical conjecture and extends the characteristic-zero relationship between generalized blob-algebra decomposition numbers and Kazhdan–Lusztig polynomials to positive characteristic.