Is the two-powerful-number representation function n^{o(1)}? (Erdős #943)

Let $A$ be the set of powerful numbers: positive integers $n$ such that $p\mid n \Rightarrow p^2\mid n$ for every prime $p$ (so $A=\{1,4,8,9,16,25,27,32,36,\ldots\}$). Let $1_A\ast 1_A(n)=\#\{(a,b): a,b\in A,\ a+b=n\}$ be the additive representation function counting ordered ways to write $n$ as a sum of two powerful numbers. Is it true that $$1_A\ast 1_A(n)=n^{o(1)}?$$ That is, does the number of representations of $n$ as a sum of two powerful numbers grow slower than any fixed positive power of $n$: for every $\epsilon>0$ is there a constant $C_\epsilon$ with $1_A\ast 1_A(n)\leq C_\epsilon n^\epsilon$ for all $n$?

Acceptance. FULLY RESOLVES: a complete proof that $1_A\ast 1_A(n)=n^{o(1)}$ (for every $\epsilon>0$ a $C_\epsilon$ with $1_A\ast 1_A(n)\leq C_\epsilon n^\epsilon$ for all $n$) — machine-checkable (Lean/Coq, e.g. extending the existing formalisation) preferred, else a full written proof; OR a disproof exhibiting an explicit infinite family $n_k$ and a constant $c>0$ with $1_A\ast 1_A(n_k)\geq n_k^{c}$, together with proof. ADVANCES (each must strictly beat what is stated in the background): (a) an unconditional upper bound $1_A\ast 1_A(n)\leq n^{\theta+o(1)}$ for an explicit constant $\theta<1/2$ (below the trivial exponent), with proof; (b) a reproducible search establishing a record lower bound — an explicit $n$ (with the list of its powerful summand pairs) achieving $1_A\ast 1_A(n)$ larger than any previously exhibited value, delivered with the enumeration program and a certificate of completeness of the representation count. Deliver the proof, or the search code plus the record $n$ and its full representation list.