Polignac's conjecture on prime gaps

Let $\mathbb{P}$ denote the set of primes, and let $\mathbf{POL}$ be the set of even numbers that occur infinitely many times as differences of two consecutive primes. Polignac's conjecture. Every even number can be written in infinitely many ways as the difference of two consecutive primes. Equivalently, $\mathbf{POL}=2\mathbb{N}$. The source notes that bounded-gap results prove only that $\mathbf{POL}\cap[2,246]\neq\emptyset$, so the full conjecture remains open.