Conjectured non-derivability from ZFC + I_1

Prove that ZFC + I_1 does not prove the consistency of the theory comprising ZF−_j plus a cofinal elementary embedding j: V → V, together with DC_λ and the existence of V_{λ+1} where λ = sup_{n<ω} j^n(crit(j)).

Background

Given the anticipated large cardinal consequences of the choiceless cofinal embedding framework (as conjectured by the author), it is suggested that the consistency of this theory is not provable from ZFC + I_1. Establishing this would clarify the relative position of choiceless cofinal Reinhardt embeddings within the large cardinal hierarchy.