Main trace conjecture for genuine Picard modular forms

Let $\lambda\neq(0,0,0)$, let $\nu$ and $\mu$ be as in the paper, and write $\lambda=(a,b,i)$ so that $i\equiv_2 a+b$. Let $T(\nu)$ be the Hecke operator, $F_\nu$ the corresponding Frobenius, $S_{n(\lambda)}^{\rm gen}(\Gamma[\sqrt{-3}])^\mu$ the $\mu$-isotypic genuine subspace, and $e_{c,\mu}(\lambda)$ and $e_{\rm extr,\mu}(\lambda)$ the cohomological terms defined in the paper. The main trace conjecture.

$$\operatorname{Tr}\bigl(T(\nu),S_{n(\lambda)}^{\rm gen}(\Gamma[\sqrt{-3}])^\mu\bigr)=\operatorname{Tr}\bigl(F_\nu,e_{c,\mu}(\lambda)-e_{\rm extr,\mu}(\lambda)\bigr).$$

This is intended to compute genuine Hecke traces from point counts after subtracting Eisenstein and lift contributions; its status is explicitly conjectural in the source.