Generalized Jørgensen trace inequality under displacement hypotheses

Let $\Gamma=\langle\xi_1,\xi_2,\dots,\xi_n\rangle$, and let $\alpha_n$ be as described in the generalized Jørgensen conjecture. For $i\ne1$, define

$$\Phi_n=\Gamma_{\o}^n-\{\xi_1,\xi_1^{-1},\xi_i^{-1}\xi_1\xi_i,\xi_i^{-1}\xi_1^{-1}\xi_i,\xi_i\xi_1\xi_i^{-1},\xi_i\xi_1^{-1}\xi_i^{-1}\}.$$

Let $z_1$ and $z_2$ be the midpoints of the shortest geodesic segments connecting the axes of $\xi_1$ to the axes of $\xi_i\xi_1\xi_i^{-1}$ and $\xi_i^{-1}\xi_1\xi_i$, respectively. Assume that there exists an isometry $\xi_i$, with $i\ne1$, such that

$$d_{\xi_i\xi_1\xi_i^{-1}}z_2\leq d_{\xi_i\xi_1\xi_i^{-1}}z_1$$

and

$$d_\gamma z_2<\tfrac{1}{2}\log\alpha_n$$

for every isometry $\gamma\in\Phi_n$. Generalized Jørgensen trace conjecture. Under these assumptions,

$$|\operatorname{trace}^2(\xi_1)-4|+|\operatorname{trace}(\xi_1\xi_i\xi_1^{-1}\xi_i^{-1})-2|\geq2\sinh^2\left(\tfrac{1}{4}\log\alpha_n\right).$$

This is presented as a consequence of the preceding conjectural displacement bound and would generalize the corresponding two-generator trace inequality. The source explicitly says that the displacement conjecture and analogous arguments imply this result.