Vertical-to-horizontal trace essential-surjectivity conjecture

Let $\operatorname{vTr}(\operatorname{SBim})$ and $\operatorname{hTr}(\operatorname{SBim})$ denote the vertical and horizontal traces of the category of Soergel bimodules, and let $\operatorname{Kar}$ denote Karoubi completion. Trace essential-surjectivity conjecture. The natural fully faithful functor

$$\operatorname{Kar}(\operatorname{vTr}(\operatorname{SBim}))\hookrightarrow\operatorname{Kar}(\operatorname{hTr}(\operatorname{SBim}))$$

is essentially surjective up to grading shifts. This would identify the Karoubi-completed vertical and horizontal trace categories, modulo grading shifts, despite the preceding observation that the corresponding bounded homotopy functor need not be essentially surjective outside type $A$. The paper presents this as an expectation and gives no proof or disproof.