Categoricity at I0 cardinals

Determine whether, for an I0 cardinal λ (i.e., a singular cardinal of cofinality ω admitting an elementary embedding j: L(V_{λ+1}) → L(V_{λ+1}) with critical point below λ), every finite complete second‑order theory with a model of cardinality λ is categorical, or at least categorical among its models of cardinality λ.

Background

For I0 cardinals, λ is singular of cofinality ω, measurable in L(V_{λ+1}), and Choice fails in L(V_{λ+1}). The failure of AC implies there is no second‑order definable well‑order of P(λ), hindering standard categoricity arguments that use definable well‑orders.

This problem asks whether categoricity nonetheless holds for finite complete second‑order theories with models of cardinality λ of an I0 cardinal, at least within the class of models of that fixed cardinality.