A positive relative-density set $A$ with all $n-a$ prime for infinitely many $n$ (Erdős #428)

Let $\pi(x)$ denote the number of primes up to $x$. Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, the integer $n-a$ is prime for every $a\in A$ with $0<a<n$, and moreover $$\liminf_{x\to\infty}\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?$$ The density of $A$ is here measured against the primes through the prime-counting function $\pi(x)$, so the condition asks $A$ to remain a positive proportion as large as the set of primes, in the liminf sense, while its elements (together with $n$) form a prime constellation shifted below $n$.

Acceptance. FULLY RESOLVES: a complete rigorous proof — a Lean/Coq formalisation matching the DeepMind formal-conjectures statement for #428 is preferred, otherwise a full written proof — that either constructs such a set $A$ (establishing both the infinitely-many-$n$ prime condition and $\liminf |A\cap[1,x]|/\pi(x)>0$) or proves that no such $A$ exists. ADVANCES: prove the $\liminf$ statement conditionally on the prime $k$-tuples conjecture, strengthening the known conditional $\limsup$ result stated in the background; or establish a quantitative positive lower bound on the achievable liminf density under a clearly stated hypothesis; or exhibit, by computational search, explicit admissible sets $A$ together with witnesses $n$ realising a record value of $|A\cap[1,x]|/\pi(x)$, supplying the construction and a certificate that every $n-a$ is prime. Deliver the proof artifact (formal or written) or the verified construction with its attained density ratio.