Bound the number of consecutive powerful pairs up to $x$: is it $(\log x)^{O(1)}$? (Erdős #365)

A positive integer $n$ is *powerful* (equivalently $2$-full) if every prime $p$ dividing $n$ satisfies $p^2\mid n$. A *consecutive powerful pair* is an $n$ such that both $n$ and $n+1$ are powerful; the smallest is $(8,9)$, and these $n$ are catalogued as OEIS A060355. Erdős asked two questions. First, must every consecutive powerful pair arise from a Pell-type equation — equivalently, must one of $n$, $n+1$ be a perfect square? Second, and the part that remains open: is the number of $n\leq x$ with $n$ and $n+1$ both powerful bounded by $(\log x)^{O(1)}$? That is, does there exist a constant $C$ with $$\#\{\,n\leq x : n\text{ and }n+1\text{ both powerful}\,\}\leq (\log x)^{C}$$ for all sufficiently large $x$? The first question is now known to have a negative answer (Golomb; Walker), so the live problem is this counting bound.

Acceptance. FULLY RESOLVES: either (a) a complete proof that $\#\{n\leq x : n, n+1\text{ both powerful}\}\leq (\log x)^{C}$ for some explicit constant $C$ and all sufficiently large $x$ — a machine-checkable (Lean/Coq) proof is preferred, otherwise a full written proof with every step — or (b) a disproof showing the counting function exceeds $(\log x)^{C}$ for every fixed $C$ (again with a complete proof). ADVANCES (each independently checkable): prove an unconditional upper bound on the counting function strictly sharper than any stated in the background, with proof; OR establish the $(\log x)^{O(1)}$ bound conditional on a clearly stated hypothesis (e.g. the abc conjecture) with a complete conditional proof; OR extend the exhaustive enumeration of consecutive powerful pairs (OEIS A060355) to a new record height with the search program and a reproducible certificate that the search was exhaustive up to that bound, reporting the empirical growth of the counting function. Deliver the proof file, the conditional proof, or the search code plus the attained bound and the enumerated pairs.