An Iran war that was already proving quite unpopular with the American people has entered a new, more problematic phase. That comes with the news that a US fighter jet was shot down over Iran.

There remains a lot we don’t know, including the status of the two crew members. While CNN has reported one of them has been rescued and is receiving medical treatment, we don’t know the fate of the other. This is, so far, an isolated incident that does not mean Iran is suddenly on anything close to an equal footing militarily. And there have thus far been limited American casualties, including no known deaths in the last three weeks.

But in a conflict in which military dominance is the US’ chief advantage, this episode underscores the perils of asymmetric warfare, the costs of which the American public already isn’t buying.

These events also puncture the Trump administration’s claims about its complete dominance of the skies over Iran — along with the veneer of impenetrability it has attempted to construct over the past month.

Those claims had already been contradicted in a number of cases. But this is a case in point.

President Donald Trump and Defense Secretary Pete Hegseth have suggested the United States and Israel had something amounting to free rein to fly over Iran. They cast Tehran as having no ability to counteract that.

At a March 4 briefing — nearly a full month ago — Hegseth said that such dominance was just around the corner.

“Starting last night, and to be completed in a few days in under a week, the two most powerful air forces in the world will have complete control of Iranian skies,” Hegseth said. He called it “uncontested airspace.”

“And Iran will be able to do nothing about it,” he added.

Trump has also played up this air dominance over the past two weeks.

“And we literally have planes flying over Tehran and other parts of their country; they can’t do a thing about it,” he said on March 24. He added that the United States could strike a power plant, and “they can’t do a thing about it.”

The president has said for weeks that Iran had “no navy,” “no military,” “no air force” and “no anti-aircraft systems.” In a White House address Wednesday night, he said he could hit Iran’s oil facilities, “and there’s not a thing they could do about it.

“They have no anti-aircraft equipment. Their radar is 100% annihilated,” Trump said. “We are unstoppable as a military force.”

Again, this is an isolated incident.

The administration has occasionally emphasized that there would be setbacks, including losses of life. And Hegseth in the same March 4 briefing acknowledged situations in which “a few drones get through or tragic things happen.”

But the administration’s claims about its military dominance in the skies have been absolutist, with phrases like “complete control” and “uncontested airspace,” even casting Iran as not even having the weaponry necessary to respond.

And it’s merely the latest example of Trump and those around him apparently exaggerating military success.

After the strikes on Iran’s nuclear facilities last June, Trump repeatedly said the country’s nuclear program had been “obliterated” and cast it as irretrievable. Except that’s not what an early US intelligence assessment showed. And sure enough, just nine months later, the administration again suddenly pitched Iran as an imminent nuclear threat.

Shortly after the war began, Trump falsely blamed Iran for a strike on an elementary school that we later learned was likely struck by the United States, according to a preliminary investigation and other evidence.

And just a day ago, CNN reported that Trump’s claims about the destruction of Iran’s missile launchers had been greatly exaggerated — and that the Islamic Revolutionary Guard Corps still retains about half of its capabilities.

The political problem with all of this is that the US military success is supposed to be the main thing that the administration has going for it.

Americans have little faith in the mission. They don’t think it’s been explained. The list of four objectives has constantly shifted. And perhaps the biggest problem is economic pessimism resulting from the effective closure of the Strait of Hormuz and the subsequent rising gas prices. Americans just don’t think the war is worth the costs.

Through it all, Hegseth in particular has argued that the media has given short shrift to the military success of the campaign.

“This is what the fake news misses,” he said in that same March 4 briefing. “We’ve taken control of Iran’s airspace and waterways without boots on the ground.”

A month later, the most crucial waterway remains a very important exception. And the control of Iran’s airspace and the demise of its missile launching program aren’t looking as complete as advertised.