The law most likely does not apply unless the governor is sworn into federal office. Rick Scott ran for senate in 2018 as governor, and nobody cared, including Bill Nelson. They might as well clear any ambiguity if Trump is just going to use it as a distraction. He lied about Obama's birth certificate for years. He lied about sending investigators to Hawaii. Doesn't matter ... most of the trumpers to this day would tell you Obama was born in Kenya.