Erdős's conjecture on the smallest prime divisors of Sierpiński sequences

Let $k$ be a Sierpiński number, meaning that $k\cdot 2^n+1$ is composite for every positive integer $n$. For each positive integer $n$, let the smallest prime divisor of $k\cdot 2^n+1$ be the least prime dividing that integer. Erdős's conjecture. The smallest prime divisor of $k\cdot 2^n+1$ remains bounded as $n$ tends to infinity. This conjecture concerns whether the prime divisors witnessing the compositeness of a Sierpiński sequence can always be chosen from a bounded set; the supplied text gives no resolution, so its status is open.