The Grothendieck-group realization conjecture for the affine Hecke trace

Let $\mathcal{A}$ be the algebra generated by the elements $R_{\mathbf d}$ over $\mathbb{Z}[q_1^{\pm1},q_2^{\pm1}]$, and specialize its parameters by

$$(q_1,q_2)=(q^{-2},q^2).$$

Let $G(\operatorname{Tr}(\mathrm{ASBim}_n))$ denote the Grothendieck group of the trace of the affine Soergel-bimodule category in rank $n$. Grothendieck-group realization conjecture. There is an isomorphism $\mathcal{A}\big|_{(q_1,q_2)\to(q^{-2},q^2)}\cong\bigoplus_{n=0}^{\infty}G(\operatorname{Tr}(\mathrm{ASBim}_n)).$ The algebra $\mathcal{A}$ is introduced because its PBW-type generators and relations are expected to model the Grothendieck groups of the affine Hecke traces; the source does not provide a resolution of this isomorphism.