If a conspiracy occurred, it was 50 years ago when Obama's grandmother used her influence to get his vital records admitted. At the time, it would have been a minor event lost among the routine mundane affairs of a local bureaucracy. To those in the future who look back at it, it would appear at first glance to be in order. To those in the past, it would have been like fixing a parking ticket, nothing to brag about and easy to forget or minimize in thought.
Any conspiracy in the current would be from those who are familiar with the bureaucracy and the normal state of documents in the entirety, and who can then recognize anomolies such as the absence of related documents that should also be present. Theirs is a passive consipiracy, as they were not a part of the falsification of records, but they are silent about inconsistencies found in the records.
-PJ