Conjectural normalized Euler characteristic formula for genus 2

Let $k>0$ and $l\geq0$ satisfy $k\equiv_6 l$. Let $\mathcal X^{(2)}_{\Gamma[\sqrt{-3}]}$ be the genus-2 moduli space considered in the paper, let $\mathbb W_{k,l}$ be the corresponding local system, and let $\alpha_k$, $\beta_k$, $S^\pm[\Gamma_1(3),k+2]$, $\mathbb L^{r,s}$ and $\delta_i$ be the representations and spaces defined in the surrounding text. The genus-2 normalized Euler-characteristic conjecture.

$$\begin{aligned} e_c^{\rm norm}(\mathcal X^{(2)}_{\Gamma[\sqrt{-3}]},\mathbb W_{k,l})={}&-\alpha_k-\beta_k-S^+[\Gamma_1(3),k+2]\alpha_k-S^-[\Gamma_1(3),k+2]\beta_k\\ &+\delta_{k+1}(\mathbb L^{k+1,0}\alpha_k+\mathbb L^{0,k+1}\beta_k)\\ &+\delta_{k+7}(\mathbb L^{0,k+1}\alpha_k+\mathbb L^{k+1,0}\beta_k), \end{aligned}$$

as an element of $K^{\mathfrak S_2\times\mathfrak S_2}_0(\operatorname{Gal}_F)$, where $\delta_i=1$ if $i\equiv_{12}0$ and $0$ otherwise. This supplies a genus-2 analogue of the preceding cohomological conjectures.