Density-one 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. Let $\Phi_{n,f}$ denote the period-$n$ dynatomic polynomial of $f$, and let $\mathcal S_{\Phi_{n,f}}$ be the set of primes at which the relevant local period-$n$ condition holds; write $\delta(\mathcal S_{\Phi_{n,f}})$ for its density. Density-one local-global conjecture. One has

$$\delta(\mathcal S_{\Phi_{n,f}})<1.$$

This is a stronger formulation of the local-global principle considered in the paper: the relevant period-$n$ condition should fail on a positive-density set of primes. The statement is proved in the paper under the associated period-$5$ rational-point hypothesis, but remains open in general.