This is a big week for the Bush family as the opening of George W. Bush’s presidential library and museum on the campus of Southern Methodist University has brought the 43rd president’s legacy into focus. The debate over his record has been fierce but, as Peter Wehner noted yesterday a Washington Post-ABC News poll gave Bush supporters some long-needed comfort as it showed his approval rating was roughly equivalent to that of his successor. Some are interpreting this result as an indicator that the day Republicans had waited for had finally arrived as the public finally realizes Bush’s worth while catching on to Barack Obama’s shortcomings.

The GOP celebration may, however, be a bit premature. One poll does not constitute a trend and one would think that the last presidential campaign would have cured Republicans of their habit of placing their faith in polls that produced results that pleased them. The timing of the survey, which was taken last week in the immediate aftermath of the Boston Marathon bombing, may also have influenced the numbers as it highlighted the one issue—homeland security and terrorism—on which President Bush always scored relatively well even when his popularity was its nadir.

Let’s assume for the sake of argument that what we’re seeing in the WaPo poll is at least the beginning of a shift in public opinion about Bush 43. As I’ve written before, the opprobrium with which his presidency has been treated since he left office is largely undeserved. He made his share of mistakes but, as Bush supporters are pointing out this week, his defense of the homeland after 9/11 was his greatest achievement and the keynote of his presidency. If the worm is turning on Bush, this might mean the path is clearing for a third member of the family to try for the White House. That’s the conceit of much of the recent coverage of Jeb Bush, whose obvious interest in a 2016 run is also being highlighted by the big party in Dallas. But any assumptions that the uptick in his brother’s poll numbers mean that there is no Bush fatigue in the country are probably unfounded.

As former Republican Party chair Haley Barbour told Politico today, the calculations about Jeb’s presidential hopes are inextricably tied up with the whole notion of Bush fatigue. Barbour is probably right when he says, “If Jeb’s last name was Brown instead of Bush, he’d probably be the front-runner for the Republican nomination.”

As a successful former governor of crucial state with a strong conservative record and a history of appealing to Hispanics, he fits the profile of exactly what the GOP is looking for in 2016. Even more than that, as one of the party’s most thoughtful voices on issues like education and immigration, he’s well prepared to make a strong case for himself as someone linked to the party’s future rather than its past.

But as Barbour says, Jeb’s name is Bush, not Brown. And his belief that there is no such thing as Bush fatigue is profoundly mistaken.

No matter how qualified Jeb Bush may be, Republicans understand that, like it or not, his presidential candidacy would inevitably become a referendum on his family’s place in American history. His own statements, both this year and last, defending his brother make it abundantly clear that the issue will follow him around wherever he goes even if he wants to talk about everything else.

Bush fatigue may be declining as the years pass and Bush 43’s accomplishments are recognized and Katrina, the Iraq War and the financial meltdown are no longer in the news. But a Jeb Bush candidacy will serve as an excuse for the left and the media to double down on their past attacks rather than allowing them to fade from our collective memory. Anyone who thinks the same elements that largely control the mainstream media and popular culture that buried the second President Bush under an avalanche of vituperation are not prepared to renew their attacks is underestimating the hatred that he engendered on the left.

During a week when George W. Bush is finally getting a little credit after years of being wrongly slammed as the man who lied us into war and crashed the economy, it may be possible for his family to dream of an unprecedented presidential trifecta. But Republicans should be wary of their ambitions. No matter how strong their arguments about Bush 43’s virtues, one poll doesn’t change the fact that his presidency is still associated with a hurricane, missing weapons of mass destruction, a bloody and inconclusive war and the bailout of Wall Street as the economy tottered. Sadly, Bush fatigue is not a figment of a hostile media’s imagination. GOP hopes in 2016 depend on convincing the American people their party is the hope of the American future after eight dismal years of Obama. Another Bush candidacy is a recipe for GOP disaster.