The Karoubi-generation conjecture for affine Soergel trace categories

Let $n\in\mathbb{N}$ and let $\operatorname{Tr}(\mathrm{ASBim}_n)^{\operatorname{Kar}}$ be the idempotent completion of the trace category of affine Soergel bimodules. For integers $m_i$ and positive integers $n_i$ satisfying $n_1+\dots+n_k=n$, order the pairs by slope $m_1/n_1\leq\dots\leq m_k/n_k$, and let $\bar E_{m_i,n_i}$ denote the corresponding $S_d$-antisymmetric objects. Karoubi-generation conjecture. The category $\operatorname{Tr}(\mathrm{ASBim}_n)^{\operatorname{Kar}}$ is generated by

$$\left\{\bar E_{m_1,n_1}\star\dots\star\bar E_{m_k,n_k}\right\}^{\frac{m_1}{n_1}\leq\dots\leq\frac{m_k}{n_k}}_{\substack{(m_i,n_i)\in\mathbb{Z}\times\mathbb{N},\\ n_1+\dots+n_k=n}}.$$

Here generation means that every object is homotopy equivalent to a chain complex built from these objects and their grading shifts. The conjecture concerns whether the expected generators of the non-idempotent-complete trace category continue to generate after Karoubi completion; the supplied source also notes that the analogous geometric functor is expected to send these objects to their commuting-stack counterparts.