It will go nowhere because of past mistakes made by the Judiciary, specifically Prigg v. Pennsylvania, which ruled that State officials didn't have to enforce Federal (and in this case specifically constitutional) Law.
As to the authority so conferred upon state magistrates [to deal with runaway slaves], while a difference of opinion has existed, and may exist still on the point, in different states, whether state magistrates are bound to act under it; none is entertained by this Court that state magistrates may, if they choose, exercise that authority, unless prohibited by state legislation.
This thing opened the door for States refusing to enforce constitutional requirements.
Hawaii Health Bureau statistical data for the Obama birth was published on August 13, 1961 in the Honolulu Advertiser newspaper. If that 1961 data was fake, I can’t imagine anyone holding anyone from 2007 to 2017 responsible for it.
Newspaper announcements do not prove that someone was born in a state when that state law allows people born elsewhere to get a birth certificate from that state so long as their parent was a resident.
In *NORMAL* states, it would likely be sufficient, but Hawaii is different. It has a system that allows births outside the state to be recorded as having occurred inside the state. This is a fact that you have no interest in addressing other than to say "They wouldn't do that."
We do not know what they would do. I have read plenty enough articles alleging a US Citizenship mill for the entire Pacific Rim through Hawaii's lax laws on the issuance of birth certificates.