[FOM] Question about Truthfulness of Large Cardinal Axioms

[Dmytro Taranovsky]

Are there set theorists who believe that

I3: For some ordinal lambda, there is a nontrivial elementary embedding of
V_lambda into itself

is consistent with ZFC but false?  If so, what is their reasoning?

An argument for negation of I3 would be an intuitively true combinatorial
principle that is inconsistent with I3.  For example, the prevailing view is
that "ZF + for a regular kappa, there is a nontrivial elementary embedding of
V_kappa into itself" is consistent but false: We accept the axiom of choice and
thus reject the quoted theory.  However, I see no sign of a natural
strengthening of the axiom of choice that is inconsistent with I3.

Sincerely,
Dmytro Taranovsky