Does the generalized continuum hypothesis entail {\diamondsuit(E^{\lambda^+}_{\operatorname{cf}(\lambda)}}) for every singular cardinal…

Does the generalized continuum hypothesis entail {\diamondsuit(E^{\lambda^+}_{\operatorname{cf}(\lambda)}}) for every singular cardinal \lambda?

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Source: List of unsolved problems in mathematics (CC BY-SA 4.0)