As I’ve been noting here, a key danger for Obama is that swing voters appear open to the basic premise of Mitt Romney's candidacy: that his experience turning around troubled companies has left him well equipped to turn around an entire country and its economy.

A recent Gallup poll found that 55% say the economy would get better over the next four years if Romney were elected, compared with 46% who say it would improve if Obama were re-elected — suggesting voters already may equate Romney’s business success with the economic competence they seem to want in a president.

On a conference call with reporters just now, I asked Obama campaign adviser Stephanie Cutter to respond to these numbers — and asked whether they are worrisome to the Obama team. Cutter acknowledged that the campaign has more work to do to persuade voters to see Romney’s business experience on the Obama campaign’s terms.

“I think what that shows is that nobody has a good understanding of what Romney’s economics are or what Romney’s business experience really was,” Cutter said, adding that in a “rare moment of candor” Romney recently admitted that his experience was “about wealth creation.”

“What we’re doing today and what we’re doing all week is explaining that wealth creation came at a great cost to middle class communities,” Cutter continued.

Cutter added that Romney’s record as Governor of Massachusetts — which ranked 47th in job creation — would be central to the case against Romney’s economic vision and ideas, along with his record as a “corporate buyout specialist.”

“Over time, people will become aware of these things,” Cutter said. "To the extent that Bain was introduced in the Republican primary, consistently Mitt Romney lost the middle class vote. They were introduced to what his business experience really meant for middle class families.”

The key takeaway here: The Obama campaign is fully aware of the danger that voters are open to accepting Romney’s version of the meaning of his Bain years, and that they may grant Romney the presumption of economic competence; that highlighting his Bain record is absolutely central to undermining that aura of competence; and that the Obama team knows that winning this argument is going to take time and is pivotal to the race.

By the way: Romney is again back to making the claim that he created over 100,000 jobs at Bain. Romney has veered wildly back and forth on the number of jobs he “created” at the company, sometimes downgrading it to “thousands,” and other times putting it at 10,000. After Post fact checker Glenn Kessler pronounced the 100,000 number “untenable,” it seemed to disappear for awhile, but now the 100,000 figure appears to be back from the dead.