Convex-gap prime sequences: must $q_n/n^2\to\infty$? (Erdős #455)

Let $q_1<q_2<\cdots$ be an infinite increasing sequence of primes whose consecutive gaps are non-decreasing: $$q_{n+1}-q_n\geq q_n-q_{n-1}\quad\text{for all }n.$$ Must it then be the case that $$\lim_{n}\frac{q_n}{n^2}=\infty\,?$$ Here the $q_n$ need not be all primes in order, only an increasing sequence of primes satisfying the stated convexity (non-decreasing gap) condition.

Acceptance. FULLY RESOLVES (proof-shaped): a complete proof — machine-checkable (Lean/Coq) preferred, else a full written proof — that every increasing sequence of primes with non-decreasing gaps satisfies $\lim_n q_n/n^2=\infty$; OR a proof of the negation, i.e. a construction of such a sequence (with proofs of its convexity, its infinitude, and that its members are prime) for which $\liminf_n q_n/n^2<\infty$. ADVANCES: strictly improve the best known lower bound on $\liminf_n q_n/n^2$ beyond Richter's value stated in the background, with a proof or reproducible certificate; or prove $\liminf_n q_n/n^2=\infty$ under a clearly stated hypothesis. Deliver the proof, or the construction together with the certified bound.