We report the falsification of the following conjecture: For every integer n >= 3, if the Fibonacci number F_n is prime, then n must be a prime number, AND the index n satisfies the property that 2n+1 is either a prime number or a semiprime (product of exactly two primes, not necessarily distinct). Further. A counterexample was discovered computationally: witness = 13. This result was obtained by the SOVEREIGN autonomous research system.