Harder's conjecture and Hermitian automorphic forms

Let $k\ge4$ and $j\ge2$ be integers with $j$ even. Harder's conjecture predicts a congruence between a primitive elliptic cusp form of weight $2k+j-2$ and a degree-two vector-valued Siegel Hecke cusp eigenform of weight ${\det}^{k}\mathrm{Sym}^{j}$. We prove the corresponding Harder-type congruence