Strong local-global conjecture for periodic points of quadratic polynomials

Let $f\in\mathbf Q[x]$ be a quadratic polynomial, and let $n>3$ be an integer. For a prime $p$, write $\mathbf Q_p$ for the field of $p$-adic numbers. A point has period $n$ for $f$ if its forward orbit under iteration by $f$ has exactly $n$ distinct points. Strong local-global conjecture. There exist infinitely many primes $p$ such that $f$ does not have a point of period $n$ in $\mathbf Q_p$. This strengthens Poonen's conjecture by requiring the failure of a period-$n$ point locally at infinitely many primes. The claim is established for periods $4$ and $5$ in the paper's preceding theorems, while the general case remains open.