Monotonicity conjecture for translation lengths in complex hyperbolic triangle-group deformations

Let $I$ be the critical interval for the family $\rho_t(p,q,r)$, let $\Gamma$ be the abstract $(p,q,r)$ triangle group, and for an infinite word $W\in\Gamma$ write $W_t=\rho_t(W)$. Let $\lambda(W_t)$ denote the translation length of $W_t$. Translation-length monotonicity conjecture. As $t$ increases monotonically from $0$ to the boundary $\partial I$ of the critical interval, the quantity $\lambda(W_t)$ decreases monotonically for every infinite word $W$. The source places this claim in the broader conjectural picture of how representations vary as the deformation approaches the boundary of the critical interval.