Dominance threshold conjecture for random homogeneous ideals in three variables

Let $n=3$, and let $p$ be the probability that each degree-$d$ monomial is selected independently in the homogeneous random monomial ideal model. As the generating degree $d$ increases, the probability that the resulting random homogeneous ideal is dominant tends to zero whenever $p>0.1$.

Dominance threshold conjecture. As $d\to\infty$, for every fixed $p>0.1$, the probability that the random homogeneous ideal is dominant tends to $0$.

This conjecture formalizes the simulation-based observation that, for three variables, dominance disappears asymptotically above the apparent probability threshold $0.1$. The source suggests that analogous thresholds may hold for general numbers of variables, with a suitable threshold depending on $n$ and $d$, but does not state those generalizations as part of this conjecture.