Tightened Goldbach conjecture

Let $\mathbb{N}$ denote the natural numbers and $\mathbb{P}$ the primes. For each $n\in\mathbb{N}$ with $n\geq 6$, let $k_n$ be the index of the uniquely determined prime $p_{k_n}\in\mathbb{P}$ satisfying

$$p_{k_n}^2+p_{k_n}\leq 2\cdot n<p_{k_n+1}^2+p_{k_n+1}.$$

Tightened Goldbach conjecture. For all $n\geq 6$, there exist primes $q_1,q_2\in\mathbb{P}$ such that

$$p_{k_n}<q_1<p_{k_n}^2,\qquad q_1+q_2=2\cdot n.$$

The source reports verification for all $n$ with $12\leq 2\cdot n\leq 10^8$, but gives no proof for the remaining cases.