Responding to Mayor-President Sharon Weston Broome's call to fire the officer who killed Alton Sterling, Baton Rouge Police Chief Carl Dabadie wrote that any such action would be improper and premature until the criminal investigation is complete.

Dabadie's letter is the latest volley in the public back-and-forth between the mayor-president and police chief over the fate of police officers Blane Salamoni and Howie Lake II in the July shooting death of Sterling.

Broome had sent Dabadie a letter May 25 asking him to fire Salamoni, who the mayor-president said "seriously violated established police policy," as U.S. Department of Justice officials said in closed-door meetings that Salamoni held a gun to Sterling's head while using expletives and threatening to kill him.

The mayor-president's call came after attorneys for Sterling's children sent her a letter pressuring her fire Salamoni.

She gave the police chief 24 hours to respond.

Dabadie's response, dated May 26, appears to have fallen within that 24-hour window. He responded in letter form, writing that it would be "improper" and "premature" to discipline either officer while the criminal investigation is pending.

BRPD "has always allowed due process" for officers alleged of criminal conduct, he wrote. Broome, in her letter, wrote that she understood due process but she was concerned about a violation of police policies.

Justice Department officials announced in early May they would not pursue civil rights charges against either officer. Louisiana Attorney General Jeff Landry's office is now investigating whether Salamoni and Lake violated state law.

Dabadie also noted that any violation of the law by an officer could also be a violation of BRPD policy. Pre-disciplinary hearings are not held until officers are arrested, indicted or cleared in criminal investigations, he wrote.

"I feel that waiting for all of the evidence in this case to be gathered is the most prudent course of action I can take at this time," Dabadie wrote. "I urge you to reconsider your position and let the criminal process conclude."

Broome's spokeswoman, Janene Tate, confirmed that Broome's office received Dabadie's response.

Broome and Dabadie were never on the chummiest of terms. When she was campaigning for office last year she said she would replace him as police chief, and at the beginning of this year, she said she was launching the search for his replacement. But Civil Service protections have stymied her efforts, as they do not allow the Baton Rouge mayor to fire the police chief without just cause.

For a few months, both Broome and Dabadie spoke of nothing but "positive and progressive" conversations happening between them. In the week after the Department of Justice announcement in the Sterling case, Broome said she and Dabadie were "on the same page" in terms of wanting to improve the police department, while he said they were "working together to bring about changes."

But their stances on police policy began to publicly part ways after a May 22 Advocate story revealed that a Baton Rouge police officer was placed on administrative leave after allegations he sent a racist text message to other officers. Broome said the officer should be fired and that the text is indicative of the need for top-down change in the police department.

Dabadie, on the other hand, said the department will go through the process of an internal investigation.

Broome then publicly announced May 24  she wanted Salamoni fired and Lake disciplined, and made it clear Dabadie held the sole authority to do so. She said Dabadie's next steps would be important to the community, to Sterling's family and to the police department.

Follow Andrea Gallo on Twitter, @aegallo.​