Modular-9 Hecke-trace congruence for genuine forms

Let $j,k,l\geq0$, let $\nu$ be a Hecke parameter, and let $\mu\vdash4$ be a partition indexing an $\mathfrak S_4$-isotypic component. The modular-9 trace congruence conjecture.

$$\operatorname{Tr}\bigl(T(\nu),S_{j,k+3,l}^{\rm gen}(\Gamma[\sqrt{-3}])^\mu\bigr)\equiv 3\dim S_{j,k+3,l}^{\rm gen}(\Gamma[\sqrt{-3}])^\mu\pmod{9}.$$

The source presents this as an experimentally motivated congruence for genuine Picard modular forms.