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Although the D.C. Council revamped laws protecting government whistleblowers in 2010, Mr. Baumann is still critical of the difficult path those within the police department — including himself — have had to trod.

“To litigate under whistleblower laws will take well over five years and cost well over $500,000,” he said, arguing that the process needs to be streamlined further.

Relations between agency heads and labor unions representing the District’s largest public-safety agencies — the police and fire departments — have been contentious in recent years.

“And I find that tension troubling,” Mr. Mendelson said, adding that he routinely brings together parties representing labor and management for discussions.

“In the end, I can’t make parties behave better,” he said.

Using the same style of sports metaphor that Mr. Gray has a tendency to invoke, Mr. Lynch pressed that the mayor should take action in Capt. Smith’s case.

“He’s the coach, and the penalty has been called,” Mr. Lynch said. “He’s got to do something.”