First disagreement at a cardinal-correct singular cardinal

Determine whether it is consistent that a singular cardinal of uncountable cofinality at which HOD is cardinal-correct is the first ordinal at which the power sets of HOD and V disagree, namely, that Δ(HOD, V)=κ.

Background

The paper proves that if κ is singular of uncountable cofinality and 2^{cf(κ)}<κ, then κ cannot be the first point of disagreement between HOD and V. The authors explicitly ask whether the cardinal-arithmetic assumption 2^{cf(κ)}<κ is necessary, leaving open the possibility that cardinal correctness of HOD alone may permit Δ(HOD,V)=κ.