Conjectured monotonicity and asymptotics of the critical drift for the frog model on trees

Let $\operatorname{FM}(d,p,\nu)$ be the frog model on the rooted $d$-ary tree with drift parameter $p$ and initial particle-distribution parameter $\nu$. Let $V_{\operatorname{FM}(d,p,\nu)}$ denote its visited set, and let $p_d$ be the critical drift for the one-particle-per-site model. For random visited sets, write $X\preceq Y$ when $Y$ stochastically dominates $X$. Conjectured monotonicity and asymptotics.

(i) If $d\leq d'$ and $p\leq p'$, then

$$V_{\operatorname{FM}(d,p,\nu)}\preceq V_{\operatorname{FM}(d',p',\nu)}.$$

(ii) $p_{d+1}<p_d$.

(iii)

$$\lim_{d\to\infty}p_d=\frac{2-\sqrt{2}}{4}\eqqcolon q^*,$$

where $q^*\approx 0.1464$ is the critical drift for a branching random walk that doubles only when moving away from the root. These statements concern the phase transition between recurrence and transience of the frog model; part (i) is known when $d'=kd$ and $p'=p$, while the strict monotonicity and limiting value remain unresolved in the supplied text.