Separation of volume classes for dense representations of the free group

Let $F_2$ be the free group of rank $2$, let $\operatorname{PSL}_2\mathbb{C}$ be the group of orientation-preserving isometries of hyperbolic $3$-space, and let $\rho:F_2\xrightarrow{\ \ }\operatorname{PSL}_2\mathbb{C}$ be a dense representation. Write $\operatorname{vol}$ for the bounded volume class and $v_3$ for the volume of a regular ideal hyperbolic tetrahedron. For another representation $\rho':F_2\xrightarrow{\ \ }\operatorname{PSL}_2\mathbb{C}$, let $[\rho^*\operatorname{vol}]$ and $[{\rho'}^*\operatorname{vol}]$ denote the corresponding bounded-cohomology classes. Volume-class separation conjecture. If

$$\left\|[\rho^*\operatorname{vol}]-[{\rho'}^*\operatorname{vol}]\right\|<v_3,$$

then $\rho'$ is conjugate to $\rho$. The claim would imply that distinct conjugacy classes of dense representations are separated in the seminorm by their volume classes; the source presents it as an expectation, and no resolution is supplied here.