The latest Washington Post poll does give some support to those who think the significance of the Iraq issue is in decline. Still, it's pretty scant support. This poll still has Iraq essentially tied with "economy/jobs" as the most important issue in the public's mind. I think what we see here is that if the economy keeps getting worse, that really might push Iraq off the front burner.
Still, according to the Post Iraq has way more salience as an issue than does immigration, even though the press have been pestering candidates with non-stop immigration questions and downplaying the war.
Meanwhile, re-reading Beinart and Brooks I think I have a clearer sense of what they're trying to do. Both peg the declining fortunes of Rudy Giuliani and Hillary Clinton (though in her case, I'm not really convinced her fortunes actually are declining) to this alleged decline in public concern about Iraq. One might more parsimoniously attribute the declining fortunes of each party's most hawkish candidate to either coincidence or else to the declining public appeal of hawkish policies, but that wouldn't do. Instead, the hawks suffer because people don't care as much about national security. Because caring about national security is identical to being hawkish. Very Serious Stuff.