Spectral classification conjecture for hyperbolic orbifolds

Let $E\subset \mathbb{R}$ be the set of values $\ell_1(X)$ as $X$ runs over all hyperbolic orbifolds. Let $\ell_1^{(0)}\approx15.79023$ be the smallest real number such that $E\cap(\ell_1^{(0)},+\infty)$ is discrete.

Spectral classification conjecture. The set $E$ is bounded above by $44.8883537$. The only orbifolds with $\ell_1>\ell_1^{(0)}$ are isometric to $[0;2,3,7]$, $[0;2,4,5]$, $[0;3,3,4]$, or $[0;2,3,8]$. The only orbifolds with $\ell_1=\ell_1^{(0)}$ are isometric to $[0;2,3,9]$ or the $\mathbb{Z}_3$-symmetric point in the moduli space of $[0;2,2,2,3]$ orbifolds. Moreover, $E$ contains a non-empty open interval $(\ell_1^{(1)},\ell_1^{(0)})$ containing the values $\ell_1(X)$ for other points in that moduli space.

The conjecture proposes a global classification of the largest spectral values of hyperbolic orbifolds. The numerical equalities in the statement are non-rigorous estimates, while the preceding theorem supplies partial rigorous restrictions above a lower threshold.