The short-word discreteness conjecture for complex hyperbolic triangle groups

Let $C_1,C_2,C_3$ be the sides of a complex hyperbolic triangle in ${\mathbb{H}^2_\mathbb{C}}$, with consecutive sides meeting at angles $\pi/n_k$, where $n_k\geq 3$ or $n_k=\infty$ for angle $0$. Assume $n_1\leq n_2\leq n_3$, and let $I_k\in PU(2,1)$ be the complex reflection in the side opposite the angle $\pi/n_k$. Define

$$W_A=I_1I_3I_2I_3,\qquad W_B=I_1I_2I_3.$$

The short-word discreteness conjecture. A $(n_1,n_2,n_3)$-complex hyperbolic triangle group is a discrete embedding in $PU(2,1)$ if neither $W_A$ nor $W_B$ is non-elliptic.

The conjecture proposes that discreteness is determined by the ellipticity of two words of short length. It remains open, although it has been proved for ideal triangle groups, for $(n_1,n_2,n_3)$-triangle groups with sufficiently large $n_1$, and for $(3,3,n)$-triangle groups.