Goldbach-Intervals conjecture

Let $\mathcal{P}$ be the set of prime numbers, let $\mathcal{E}$ be the set of even natural numbers, and let $\mathcal{I}_n$ be the $n$-primorial interval. Define

$$\mathcal{P}_n=\mathcal{P}\bigcap\mathcal{I}_n,\qquad \mathcal{E}_n=\mathcal{E}\bigcap\mathcal{I}_n.$$

Goldbach-Intervals conjecture. For every positive natural number $n$ and every $m\in\mathcal{E}_{n+1}$, there are at least two prime numbers $p\in\mathcal{P}_n$ and $q\in\mathcal{P}_n\bigcup\mathcal{P}_{n+1}$ such that

$$m=p+q.$$

The source reports verification for the first two primorial intervals and says that the first ten intervals can be checked computationally. It also states that this conjecture would imply the binary Goldbach conjecture, but no general proof is supplied.