The infinitude of primes of weight 3

Let $p_n$ denote the $n$th prime, and let its weight be the weight in the paper's decomposition of primes. The smallest member of each twin-prime pair other than $3$ has weight $3$. Weight-three prime conjecture. The number of primes with weight equal to $3$ is infinite. This is a reformulation of the twin prime conjecture in the terminology of the paper; no proof is known.