Positive-density conjecture for exceptional Tribonacci primes

Let $\mathcal{ML}$ be the set of primes $p$ for which the Marques–Lengyel conjecture holds, and let $\mathcal{NMLR}$ be the set of primes for which the rational-center variant fails. Positive-density conjecture. Both subsets $\mathcal{ML}$ and $\mathcal{NMLR}$ are infinite; in fact, both have positive lower density among all primes. This conjecture summarizes the paper's proposed heuristic picture: infinitely many primes should exhibit each of the two contrasting valuation behaviors, but the text provides heuristics rather than a proof.