The slope-product generation conjecture for affine Soergel traces

Let $n\in\mathbb{N}$, let $(m_i,n_i)\in\mathbb{Z}\times\mathbb{N}$, and let $d_i\geq 1$. For the affine Soergel trace category $\operatorname{Tr}(\mathrm{ASBim}_n)$, write $\star$ for its product and let $\operatorname{Tr}(\mathrm{ASBim}_n)^{\operatorname{Kar}}$ denote its idempotent completion. Slope-product generation conjecture. The products

$$\bar E_{m_1d_1,n_1d_1}\star\dots\star\bar E_{m_kd_k,n_kd_k},$$

where $\sum_i n_id_i=n$ and $m_1/n_1\leq\dots\leq m_k/n_k$, generate $\operatorname{Tr}(\mathrm{ASBim}_n)^{\operatorname{Kar}}$. This is a broader slope-wise generation claim for the Karoubi-completed trace category, motivated by the explicit resolutions established earlier in the source; its status is not resolved in the supplied material.