Quantitative dynamics conjectures for the relative free factor complex

Let $(\Gamma;\mathscr A)$ be the relatively free product under consideration, let $\mathcal{F\!F}(\Gamma;\mathscr A)$ be its relative free factor complex, and let $\phi\in\operatorname{Out}(\Gamma;\mathscr A)$ be fully irreducible with expansion factor $\lambda_\phi>1$. Write $\tau^{\mathcal{F\!F}}_\phi$ for the translation length of $\phi$ on $\mathcal{F\!F}(\Gamma;\mathscr A)$.

There exist constants $A^{\mathcal{F\!F}}>0$, $B^{\mathcal{F\!F}}>0$, and $\Omega^{\mathcal{F\!F}}>0$, each depending only on $\Gamma$ and $\mathscr A$, such that the following hold.

Quantitative dynamics conjectures. First, if $\phi$ is fully irreducible with expansion factor $\lambda_\phi>1$, then

$$A^{\mathcal{F\!F}}\leq\tau^{\mathcal{F\!F}}_\phi\leq B^{\mathcal{F\!F}}\log(\lambda_\phi).$$

Second, the following conditions are equivalent: (1) $\phi$ is fully irreducible; (2) $\phi$ acts loxodromically on $\mathcal{F\!F}(\Gamma;\mathscr A)$, equivalently $\tau^{\mathcal{F\!F}}_\phi>0$; (3) $\phi$ acts with unbounded orbits on $\mathcal{F\!F}(\Gamma;\mathscr A)$; and (4) every orbit of the action of $\phi$ on $\mathcal{F\!F}(\Gamma;\mathscr A)$ has diameter at least $\Omega^{\mathcal{F\!F}}$. The equivalence of (1) and (2) is already known, while the lower bound in the first assertion and the implication (4) $\implies$ (1) for general groups $\Gamma$ remain unresolved. These conjectures seek a quantitative refinement of the qualitative loxodromic-versus-elliptic dynamics theorem for the action of $\operatorname{Out}(\Gamma;\mathscr A)$ on relative free factor complexes.