Faber–van der Geer Hecke-trace conjecture for vector-valued Siegel cusp forms

Let $a\geq b\geq 0$ with $(a,b)\neq(0,0)$, let $V_{a,b}$ be the corresponding local system on $\A{2}$, let $T(p)$ be the Hecke operator at $p$, and let $S_{a-b,b+3}$ be the space of cusp forms on ${\rm Sp}(4,{\bZ})$ of the indicated vector-valued weight. Define

$$e_{2,\rm extra}(a,b)=s_{a-b+2}-s_{a+b+4}(S[a-b+2]+1){\bL}^{b+1}+\begin{cases} S[b+2]+1 & a\text{ even},\\ -S[a+3] & a\text{ odd},\end{cases}$$

where $s_n=\dim S_n({\rm SL}(2,{\bZ}))$ and ${\bL}=h^2({\bP}^1)$ is the Lefschetz motive. Faber–van der Geer conjecture. The trace of $T(p)$ on $S_{a-b,b+3}$ equals

$$-\operatorname{Tr}(F_p,e_c(\A{2}\otimes {\bF}_p,V_{a,b}))+\operatorname{Tr}(F_p,e_{2,\rm extra}(a,b)).$$

The formula is proved for regular highest weights $a>b>0$ using work of Weissauer, while it remains open for non-regular highest weights.