"The notification further said that only grocery shops, convenience stores and pharmacies would be allowed to remain open in these areas" here why we used " would" and not " will " ??

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@Muhammadsaadanjum “would” suggests that these places have the permission and ability to remain open (and refers to the future).

To use “will” here would not be incorrect, btw. It would have essentially the same meaning. “Will” emphasizes the certainty of the future, and “would” emphasizes the ability of the future.