(This is a reformulation of an earlier unanswered question. I would like to thank Ian Agol for pointing out to me Walter Parry's characterization of hyperbolic translation length functions.)

Let $G$ be a finitely generated group and $T_1$ and $T_2$ two minimal (and hence cocompact) metric simplicial trees that $G$ acts on isometrically by simplicial automorphisms without inversions. Consider the (unbased) hyperbolic translation length functions $$l_{T_i}\colon G\to [0,\infty),\ g\mapsto\inf_{x\in T_i}d(x,gx),\ i\in\left\{1,2\right\}.$$

Assume that $T_1$ and $T_2$ have the same covolume, say covolume 1. Moreover, suppose that for all $g\in G$ we have $l_{T_1}(g)\leq l_{T_2}(g)$ and $l_{T_1}(g)=0\Leftrightarrow l_{T_2}(g)=0$.

Does this imply that $l_{T_1}$ and $l_{T_2}$ are *equal*?

I have the feeling that for every $g$ such that $l_{T_1}(g)$ is strictly smaller than $l_{T_2}(g)$ there must be an $h\in G$ such that $l_{T_1}(h)$ is strictly bigger than $l_{T_2}(h)$ in order to preserve the covolume. Indeed, I can prove the statement for free $F_n$-trees (elements of Culler-Vogtmann Outer space), but my arguments do not generalize immediately.