Fenchel–Nielsen parameter conjecture for Zariski-dense free subgroups of ${\rm SU}(3,1)$

Let $\langle A,B\rangle$ be a Zariski-dense free subgroup of ${\rm SU}(3,1)$ generated by loxodromic elements $A$ and $B$. Let $\operatorname{tr}(A)$ and $\operatorname{tr}(B)$ denote their traces, let $\sigma(A)$ and $\sigma(B)$ denote the corresponding loxodromic invariants, let $\mathbb X_k(A,B)$ for $k=1,2,3$ denote the three cross-ratio invariants, and let an $\eta$-invariant and a $\nu$-invariant be chosen as in the preceding parameterization.

Fenchel–Nielsen parameter conjecture. The subgroup $\langle A,B\rangle$ is determined up to conjugacy by

$$\operatorname{tr}(A),\ \operatorname{tr}(B),\ \sigma(A),\ \sigma(B),\ \mathbb X_k(A,B)\ (k=1,2,3),\ \text{one }\eta\text{-invariant, and one }\nu\text{-invariant}.$$

This conjecture proposes that these trace, loxodromic, cross-ratio, and angular-type invariants form a complete set of parameters for the conjugacy class of a Zariski-dense free subgroup generated by two loxodromic elements. The source presents it as a general expectation, and no resolution is supplied here.