I thought to myself 'well, it is Arkansas!' Unfortunately, the same link mentions that similar laws are also on the books in Tennessee and...Texas. Ouch! Volokh has a deeper dive on the same topic. There, I learned a bit more about the Texas case: it's in the state constitution! The official link for the Texas Constitution shows Article I (Bill of Rights), Section 4 (Religious Tests) as:
Sec. 4. RELIGIOUS TESTS. No religious test shall ever be required as a qualification to any office, or public trust, in this State; nor shall any one be excluded from holding office on account of his religious sentiments, provided he acknowledge the existence of a Supreme Being.
Whack. I had no idea it was enshrined in our state constitution!