Dimension conjecture for genuine Picard modular forms

Let $E_c(\lambda)$ and $E_{\rm extr}(\lambda)$ be the representations obtained from the compactly supported Euler characteristic and the extraneous contributions, respectively. Let $S_{n(\lambda)}^{\rm gen}(\Gamma[\sqrt{-3}])$ be the genuine subspace. The genuine-dimension conjecture. For every $\lambda$,

$$\dim_{\mathfrak S_4}S_{n(\lambda)}^{\rm gen}(\Gamma[\sqrt{-3}])=\frac{1}{3}\bigl(E_c(\lambda)-E_{\rm extr}(\lambda)\bigr).$$

This is the dimension-level consequence of the proposed cohomological decomposition and is used in the paper to predict genuine spaces and their multiplicities.