Williamson's prime-factor conjecture implying Goldbach's conjecture

Let $p_{1},\ldots,p_{k}$ denote the first $k$ odd primes in succession, and let $n$ be an even integer such that $n>p_{k}$. Williamson's prime-factor conjecture. If, for all $1\leq i\leq k$, $n-p_{i}$ is not prime, then there is an integer $i$ with $1\leq i\leq k$ such that $n-p_{i}$ has a prime factor $p$ satisfying $p\geq p_{k}$. The conjecture is presented as an elementary statement whose truth would imply Goldbach's conjecture; the supplied text gives no resolution.