First failure of covering at an uncountable-cofinality singular cardinal

Determine whether a singular strong limit cardinal of uncountable cofinality can be the first cardinal κ at which the pair (HOD,V) fails to satisfy the κ^+-cover property at κ.

Background

The paper proves that, under cardinal correctness of HOD through κ+ and GCH in HOD below κ, such a κ cannot be the first point of failure of the relevant covering property. The authors explain that a positive answer would arise if the preceding question about power-set cardinalities had a negative consistent answer, and note that GCH would then have to fail at multiple cardinals below κ.

References

Suppose that $\kappa$ is singular strong limit of uncountable cofinality. Is it possible for $\kappa$ to be the first cardinal $\lambda$ such that $(\HOD, V)$ fails to satisfy $(\lambda)+$-cover at $\lambda$?

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