Marklof's gap conjecture for arithmetic hyperbolic 3-orbifolds

Let $\Gamma_\mathfrak{D}$ be an arithmetic quaternion group. The complex length spectrum of $\mathbb{H}^3/\Gamma_\mathfrak{D}$ has gaps, and let $\mathcal{G}(x)$ denote the number of these gaps up to length $\ell=\log x$. Here $K$ is the field of definition, with degree $d=[K:\mathbb{Q}]$, and $D_\mathfrak{a}$ is the discriminant of $\mathfrak{a}=\operatorname{tr}\mathfrak{D}$. Marklof's gap conjecture. As $x\to\infty$,

$$\mathcal{G}(x)=\kappa x+o(x),$$

where $\kappa\geq 0$ depends only on $\Gamma_\mathfrak{D}$ and is small compared to $2^{2d-3}\pi |D_\mathfrak{a}|^{-1/2}$. It is possible that $\kappa=0$ for every $\Gamma_\mathfrak{D}$. This conjecture gives a conditional description of the gaps in the complex length spectra of compact arithmetic hyperbolic $3$-orbifolds; the source does not provide a resolution, and explicitly allows the possibility that the asymptotic constant vanishes.