Conjectured extension of Laver’s argument to ZF−_j with cofinal j, DC_λ, and V_{λ+1}

Develop an adaptation of Laver’s 1997 argument to the theory ZF−_j with a cofinal elementary embedding j: V → V, assuming DC_λ and the existence of V_{λ+1} where λ = sup_{n<ω} j^n(crit(j)), and demonstrate that this theory yields various large cardinal consequences such as the existence of an I_1-cardinal below λ.

Background

The author suggests that Laver’s techniques might generalize to the choiceless setting with a cofinal Reinhardt embedding, provided Dependent Choice at λ and the existence of V_{λ+1}. If this conjectured extension holds, the resulting theory would imply substantial large cardinal strength, potentially informing the relative consistency landscape for choiceless embeddings.