A new internal poll from Governor Beshear’s re-election campaign appears to confirm that the Democratic governor is running away with the race in a deep-red state while Attorney General Daniel Cameron, a close ally of Minority Leader McConnell, struggles to stay afloat. In Kentucky, though, analysts are skeptical that much has changed in the race since July.

The Kentucky gubernatorial election will be one of the premiere elections of 2023, in part because it will be a test of whether Republicans can oust a popular Democratic governor from a well-known political dynasty in an otherwise deep-red state.

A new public poll from a Democratic pollster, Garin-Hart-Yang, found that Mr. Beshear, whose father was also governor, leads Mr. Cameron by a wide margin.

The survey, obtained by the Lexington Herald Leader, found that Mr. Beshear leads Mr. Cameron 51 percent to 42 percent as of September. The same poll found that Mr. Beshear led Mr. Cameron 48 percent to 45 percent in July.

“In addition to the Governor reaching the important 50 percent vote threshold, his strength of support is an impressive 86 percent, which signifies that most Beshear voters are enthusiastic in their support for him,” the pollster wrote.

This appears to confirm other surveys taken throughout the summer, which all saw Mr. Beshear leading Mr. Cameron — at times, by as much as eight points.

The pollster also asked voters whether they thought the candidates were likable and strong leaders. Fifty-five percent saw Mr. Beshear as likable, while 28 percent said the same for Mr. Cameron. Fifty-one percent saw Mr. Beshear as a strong leader, while 35 percent said so of Mr. Cameron.

While the survey looks like it should be a warning sign for Republicans in the state, a political scientist at the University of Kentucky, Steve Voss, cautions that this shift in polling isn’t significant.

“Partly the polls underestimate Kentucky Republicans’ support because portions of Kentucky Republicans are hard to reach,” Mr. Voss said. “Another issue is that Kentucky has a lot of Republican voters who don’t think about themselves as Republicans but break in favor of Republicans.”

Mr. Voss clarified that it does look like Mr. Beshear is leading but that there is reason to be skeptical because polls have underestimated Kentucky Republicans in the past.

“Beshear appears to enjoy a healthy lead, and neither positive nor negative advertising on behalf of Cameron has been able to budge Beshear’s numbers,” Mr. Voss said. “Still, Beshear’s support hovers around the 50 percent mark, and he needs a majority to win, so if the polls are accurate, Democrats need to worry about a combination of low turnout on their side and undecideds breaking heavily Republican.”

Mr. Voss also cautioned against connecting Mr. Cameron’s lagging numbers with issues with other prominent Kentucky Republicans, such as Senator McConnell freezing up twice during public events following a concussion. “I see no reason at all to think that McConnell’s ailment has had an impact on the Kentucky governor’s race,” he said.

While Mr. Cameron has historically been touted as a successor to Mr. McConnell, Mr. Voss said Mr. Cameron’s relationship with Kentucky’s senior senator is more relevant to national political analysts and handicappers than it is to Kentucky voters.

While Mr. Cameron has enjoyed the quiet support of Mr. McConnell for years, he has also enjoyed the public support of President Trump, and Mr. Voss says his personal brand in Kentucky isn’t tied too closely with Mr. McConnell.