Harder's Conjecture and Miyawaki Lift

Let k, j and n be positive integers such that k is odd , and both j and n are even, satisfying j ≡ n mod 4. Let f and g be primitive forms of weight 2k + j - 2 and k+j/2 - n/2 - 1, respectively, for SL2(Z). Then, we propose a conjecture on the congruence between the Klingen-Eisenstein lift of the Miyawaki lift of f and g of type II and a certain lift of a vector-valued Hecke eigenform of weight (k + j, k) for Sp2(Z). This conjecture implies Harder's conjecture. Through this formulation, we prove Harder's conjecture in some cases.