If you are getting blood drawn, then there is no reason to be concerned about false negatives, unless you have an immune deficiency. The current generation of antibody test is designed to have virtually no false negatives once the person is past the window period. The current generation of lab-based antibody test is so accurate that it is now only recommended to perform testing up until 12 weeks or 3 months after exposure. There was a time that it was recommended that testing also be repeated at 6 months, but there has now been sufficient experience over years to show that testing at 12 weeks/3 months is sufficient. There has not been a seroconversion past 3 months since the current generation of test has been used, so there has not been a single case in which the test at 12 weeks/3 months has been a false negative.
The alternate methods of testing, such as finger pricks or mouth swabs, are not as accurate, and these methods may have false negative testing. But if the lab based test is performed after 12 weeks, there has not been a case of a false negative. Since no case of a late false negative has occurred, there has not been any situation that matches the question you are asking.
There is a special case that involves people that have an immune deficiency in which they do not make antibodies, which is rare, but can occur. Since these people do not make antibodies, they also will not make antibodies against HIV, and the antibody test would be negative. Actually, since there are no antibodies, this would not be considered a false negative, because the test results are accurate. It is only false in that it does not indicate whether the infection is present. This special case is not a concern for most people, because someone that has an immune system and does not make antibodies is already prone to numerous infections before they may be exposed to HIV. So, people that present for testing would already know that they have such an immune deficiency. However, in this special situation, yes, the method for detection of an HIV infection would rely on the viral load test. In this setting, then yes, the viral load test would be sufficient to be detected, even at 3 months, but would still be able to be detected at 6 months.
There are rare cases of HIV infection that are long-term non-progressors, and these people may not progress for years. The viral load test may miss these people, but these people have intact immune systems, and someone that is not able to make antibodies has not been reported to be a long-term non-progressor.
However, if you have never had an issue with frequent infections because of an immune deficiency, then there is no reason for you to be concerned about a false negative fourth generation test done after 12 weeks.
If I can provide any additional information, please let me know.