A hypothesis about the conjecture: "Every even number is the difference of two primes" If the conjecture "Every even number is the difference of two primes" holds then we conclude the following hypothesis:
Hypothesis. For every distinct non-zero integers $a, b$, at least one of the numbers $a, b$ and $a-b$ can be expressed as the difference of two primes.
Question. Is the converse true (does the hypothesis imply the conjecture)?

Yes. Take any even $n$; if it were the case that for all odd $b$, at least one of $b+2$ and $b+n+2$ were prime, the primes would have nonzero density.° So take a $b$ for which it is not the case, and the hypothesis implies $n$ is a difference of two primes.
° Partition the odd numbers in pairs with difference $n$: $\{1,n+1\}\{3,n+3\},\ldots,\{n-1,2n-1\};\; \{2n+1,3n+1\},\ldots,\{3n-1,4n-1\};\;\ldots$. There are $kn/2$ pairs below $2kn$ ($k\in\mathbb N$), so there are at least $(x-2n)/4$ pairs below $x$. Each pair has at least one prime, so $\pi(x)\geq x/8-n/4$.
