Boundedness conjecture for the $\phi_i$ parameters of positive representations

Let $\rho$ be a $\operatorname{PGL}_n(\mathbb{R})$-positive representation, and let $\mathcal{P}_\alpha$ be the collection of pairs $(\beta,\gamma)$ under consideration. For each such pair and each relevant index $i$, let $\phi_i(\beta,\gamma)$ denote the parameter defined earlier in the paper.

Boundedness conjecture. The collection

$$\{\phi_i(\beta,\gamma)\}_{(\beta,\gamma)\in\mathcal{P}_\alpha}$$

is bounded within some compact interval.

The preceding discussion establishes boundedness of the auxiliary collections entering the formulas for the parameters. The stated boundedness of the $\phi_i$ themselves is presented as a conjecture in the source, and its resolution is not supplied here.