Asymptotic length-saturation for subarithmetic hyperbolic groups

Let $\G$ be a subarithmetic hyperbolic group with growth exponent $\ga$, and let $\cO$ be the ring of integers of its trace field. The group $\G$ asymptotically length-saturates if

$$\frac{\#\cT(\G)\cap B_N}{\#\{t\in\cO\cap B_N: t\text{ is admissible}\}}\to 1$$

as $N\to\infty$, where an element $t\in\cO$ is admissible when, for every ideal $\cI\subset\cO$, it is congruent modulo $\cI$ to an element of the trace set $\cT(\G)$. Asymptotic length-saturation conjecture. If the growth exponent $\ga$ exceeds the rank of $\cO$, then $\G$ asymptotically length-saturates.