Agreement of power-set cardinalities at a singular cardinal

Determine whether, for a singular strong limit cardinal κ of uncountable cofinality such that the power-set cardinalities of HOD and V agree on a stationary set below κ and (2^κ)^HOD is a cardinal, one must have (2^κ)^HOD=2^κ.

Background

The preceding compactness theorem concerns the first disagreement between the actual power sets in HOD and V. This question weakens the agreement hypothesis by requiring only equality of the cardinalities 2λ, rather than equality of the collections of subsets. The authors ask whether stationary agreement below κ forces agreement at κ itself.

References

Suppose that $\kappa$ is a singular strong limit cardinal of uncountable cofinality, that ${\lambda < \kappa\mid (2\lambda){\HOD}=2\lambda}$ is stationary and $(2\kappa){\HOD}$ is a cardinal. Must $(2\kappa){\HOD}=2\kappa$?

— Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem  (2608.24190 - Benhamou et al., 25 Aug 2026) in Section "Open questions", Question labeled "agreement 2kappa"