Distinct-prime Goldbach variation below the largest prime

Let $n\geqslant 12$ be an even integer, and let $p$ be the largest prime less than $n$. Distinct-prime Goldbach variation. There exist distinct primes $r$ and $s$ such that

$$r<s<p\qquad\text{and}\qquad n=r+s.$$

This number-theoretic assertion is introduced because it would control the prime-graph comparison in the special cases of alternating groups discussed in the paper. It is presented as an assumed variation of Goldbach's conjecture and remains open.