The circulator trace conjecture for stable Gelfand-Kazhdan categories

Let $\Sigma$ be the set of simple objects of a stable Gelfand–Kazhdan category, with prime $p$ and lattice quotient order $L$. For $b\in\Sigma$, let

$$\sigma^{Lp+1}(b)=(1,2,3,\ldots,Lp+1):bb^{Lp}\longrightarrow bb^{Lp}$$

be the indicated circulator, and let ${\operatorname{Tr}}_{b^{Lp}\to w}$ denote the trace operation from $b^{Lp}$ to $w$.

Circulator trace conjecture. In the case of stable Gelfand–Kazhdan categories,

$${\operatorname{Tr}}_{b^{Lp}\to w}\sigma^{Lp+1}(b)= \begin{cases} 0 & \text{if }w\neq\mathbf{1},\\ \operatorname{id}_b\diamond\operatorname{id}_{\mathbf{1}} & \text{otherwise.} \end{cases}$$

The conjecture is motivated by the observation that numerically generated categories have circulator of order $Lp$ and by the computed powers of the circulator. No resolution is given in the source.