Infinite sets with $\le 2$ representations of each $n$: is $\liminf|A\cap[1,N]|/N^{1/2}=0$? (Erdős #158)

Let $A\subseteq\mathbb{N}$ be an infinite set such that, for every $n$, there are at most $2$ solutions to $a+b=n$ with $a\le b$ and $a,b\in A$ (a so-called $B_2^{+}[2]$ condition on ordered-with-repetition sums). Must $$\liminf_{N\to\infty}\frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}=0?$$ That is, does the bounded-representation constraint force the counting function of $A$ to fall below any multiple of $N^{1/2}$ along a subsequence of $N$?

Acceptance. FULLY RESOLVES: a complete proof (machine-checkable in Lean/Coq preferred, else fully written) that every infinite $A$ with at most $2$ representations of each $n$ satisfies $\liminf_{N\to\infty}\lvert A\cap[1,N]\rvert/N^{1/2}=0$; OR a construction of an infinite $A$ with the $\le 2$-representation property together with a constant $c>0$ and a proof that $\lvert A\cap[1,N]\rvert\ge cN^{1/2}$ for all large $N$, refuting the claim. ADVANCES (each with proof or reproducible certificate): prove the $\liminf$ statement under a stated extra hypothesis (e.g. for $B_2[g]$-type sets or with an averaged/Cesàro conclusion); establish a nontrivial upper bound on $\liminf \lvert A\cap[1,N]\rvert/N^{1/2}$ smaller than the trivial constant, or the analogue with $2$ replaced by a larger fixed bound; or a rigorously verified search producing dense finite initial segments of a candidate $\le 2$-representation set with the representation-count certificate, charting how large $\lvert A\cap[1,N]\rvert/N^{1/2}$ can be pushed. Deliver the written/formalised proof, or the counterexample construction, or the search code plus the verified representation-count data.