The $A_2$ character-formula conjecture for Deligne–Lusztig cohomology

Assume that ${\bf G}$ is split of type $A_2$. Let $\varphi:B^+\to\mathbb Z$ be the class function defined in the source by its values on the listed representatives and by $\varphi(\boldsymbol\pi^n{\bf b})=n+\varphi({\bf b})$. For ${\bf b}\in B^+$, let $H({\bf b})$ denote the cohomological invariant used in the source, and let $T_{\bf b}$ act on the representation $R_{-ht}$ of the relevant Hecke algebra. $A_2$ character-formula conjecture. For every ${\bf b}\in B^+$,

$$H({\bf b})=(-h)^{l({\bf b})-\varphi({\bf b})}\operatorname{Trace}(T_{\bf b}\mid R_{-ht}).$$

The claim is presented as an extension of the computed type-$A_2$ theorem and is supported by numerous further calculations in the source.