Expanded endoscopic conjecture for equal weights at level 2

Let $\mathbb V_{m,m}$ be the equal-weight local system on $\mathcal A_2[2]$, set $k=2m+4$, and write $\tau_{N,k}=\dim S_k(\Gamma_0(N))^{\rm new}$ and $\tau_k^\pm=\dim S_k^\pm(\Gamma_0(2))^{\rm new}$. Let $s[\lambda]$ denote the irreducible $\mathbb S_6$-representation indexed by $\lambda$. Equal-weight expanded endoscopic conjecture. The expanded endoscopic part is

$$L^{m+1}(L+1)\begin{cases} \tau_k^+s[1^6]+\tau_{4,k}s[3^2]+\tau_k^-s[5,1],&m\text{ odd},\\ (\tau_k^-+\tau_{1,k})s[2^3]+(\tau_k^++\tau_{1,k})s[4,2]+\tau_{1,k}s[6],&m\text{ even}. \end{cases}$$

This is the equal-weight counterpart of the expanded endoscopic prediction and accounts for the parity-dependent Saito–Kurokawa contributions.