Does "run a report" mean "release a report"?
In the sentence: They are alleged to have sought cash in exchange for not running reports on a coal scam linked to the firm. Does "running reports" here mean "releasing reports"?

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for not running reports on .......= for not  compiling  reports on...

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To run a report means to create one. That is a separate action from releasing a report.

First you would run a report, then you might or might not release it.