Hannity called Stelter a "little pipsqueak" and pointed to another segment in which a Stelter guest mused about whether Trump was a "demagogue."

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"That's the type of coverage that CNN offers in this presidential race, as they literally kiss Hillary Clinton's ass and Obama's ass every day," Hannity said.

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Stelter on his Sunday show, "Reliable Sources," spent a fair amount of time going after Hannity for the Clinton health segment. It was based on a popular conspiracy theory on conservative websites like the Drudge Report, but hasn't really penetrated the mainstream media -- mostly because it's a very serious allegation that's heavy on speculation and short on actual facts.

Drudge recently ran a banner featuring a picture of what looked like Clinton's entourage helping her up the stairs. Previously, the theories have revolved around Clinton's coughing fits and thick glasses after sustaining a concussion in a fall.

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On his show last Tuesday, Hannity asked two doctors about the possibility that Clinton might suffer from all manner of serious health problems. They even mentioned the possibility of a brain injury or a stroke. Here's part of that exchange:

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HANNITY: Is it possible she had a stroke, or do you really believe it was a head injury, traumatic brain injury? DANIEL KASSICIEH, BOARD-CERTIFIED NEUROLOGIST: I believe that she had a concussion with a possible brain injury. Of course, I haven't seen medical records, but the blood clot once it cleared up was no longer an issue. It was the latent effects of the concussion that I would be concerned about.

Fox News medical correspondent Marc Siegel said, "I saw the same video you saw, and I'm wondering about a word called 'aphasia' where you're searching for words, you suddenly lose those words, and that can be the sign, again, of some kind of traumatic brain injury or the after effects of a concussion."

Stelter said it went much too far, accusing Hannity of exploitation and rumor-mongering.

"Let me be clear: That was reckless speculation by Sean Hannity -- all of it," Stelter declared.

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He acknowledged there is a "grain of truth" behind the rumors, given Clinton's documented 2012 health problems after fainting and falling. "But Hannity is not interested in the truth about Clinton's health," Stelter said. "If he was, he could have interviewed people who were actually there during the episodes that he's exploiting."

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And then this: "Look, conspiracy theories are so much more interesting than the truth. But the last time I checked, Fox still has the word 'news' in its name."

It isn't the first time the two have tussled. After Stelter's comments on Hannity's handling of the "rigged" election story, Hannity took to Twitter to accuse Stelter of ignoring the evidence.

This has all the makings of a high-profile media feud -- and the foundation of it has been in the works for a while. Media analysts and watchdogs have long pointed to Hannity's advocacy for and softball questions to Trump and his supporters. Hannity makes little secret about his preference in the 2016 race, of course, but this isn't the first time it's crossed the line for some.

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The Washington Post's Erik Wemple even mixed it up with Hannity last month, saying it was a mistake for Fox to let him anchor its convention coverage.

Hannity shot back:

Expect more where this came from.

Update: Stelter this afternoon retweeted a brutal takedown of Hannity's medical speculation. The headline of Kurt Eichenwald's piece is, "Sean Hannity: Aploogize to Those with Epilepsy, or Burn in Hell," and Eichenwald at one point says, "F--- you, Sean."