The Constitution disagrees.
Article VI, Clause 2: This Constitution, and the Laws of the United States which shall be made in pursuance thereof; and all treaties made, or which shall be made, under the authority of the United States, shall be the supreme law of the land; and the judges in every state shall be bound thereby, anything in the constitution or laws of any state to the contrary notwithstanding.
You put in bold the very reason the given state does NOT have to follow that clause if it chooses not to. The state governor issues an executive order declaring a certain individual invalid and ineligible.
Done. Stroke of the pen, law of the land- kinda cool, huh?