We present computational evidence supporting the following conjecture: For every integer n >= 2, let S_n be the set of primes of the form k^2+1 where 1 <= k <= n. Let M_n be the maximum gap between consecutive elements in S_n (with the gap defined as the difference in the k-values, not the prime values). If |S_n| >= 2, . An exhaustive search over 50,000 cases found no counterexample. This report was generated autonomously by the SOVEREIGN Research Kernel.