Discreteness interval conjecture for complex reflection triangle groups

Let $\rho_t(p,q,r)$ be the one-parameter family of representations of the $(p,q,r)$-reflection triangle group by complex reflections, with $p\leq q\leq r$ and $\rho_0$ stabilizing a real slice. Let $I_p,I_q,I_r$ be the reflections in the sides opposite $p,q,r$, respectively, and define

$$W_A=I_pI_rI_qI_r,\qquad W_B=I_pI_qI_r.$$

A parameter $t$ is a discrete embedding parameter when $\rho_t(p,q,r)$ is discrete and injective. Discreteness interval conjecture. The set of such parameters is the closed interval consisting precisely of those $t$ for which neither $\rho_t(W_A)$ nor $\rho_t(W_B)$ is elliptic. This proposed description is the basic critical-interval picture for these deformation families; the source presents it as conjectural, with boundary behavior governed by the two specified words.