Composite values of F(2^n)+h for the Fibonacci sequence

Prove or disprove that for any integer h, the numbers F(2^n)+h are composite for infinitely many integers n, where F(n) denotes the Fibonacci sequence.

Background

To relax the strong reversibility assumption in their main theorems, the authors propose studying sequences that combine non-reversible exponential growth (2^n) with a classical linear recurrence (Fibonacci). Establishing compositeness of F(2^n)+h infinitely often would be a step toward removing the reversibility condition and could shed light on related divisibility phenomena.