No.
Because the idiots in government don’t want to do due diligence.
Also, if it is found that Brak was ineligible, that makes every law, appointment, or decree he made immediately invalid.
And they can’t have that, now.
It turns out that that is not true. The Supreme Court in 1886 established something called the "de facto officer doctrine":
"The de facto officer doctrine confers validity upon acts performed by a person acting under the color of official title even though it is later discovered that the legality of that person's appointment or election to office is deficient." - link
Of course, as hundreds of thousands of people have been researching the so-called "birther" theory for ten years, and haven't been able to come up with a single solid piece of evidence for it, that would seem to make it likely that it is just wishful thinking.