Spectral-gap conjecture for compact hyperbolic spin orbifolds

Let $X$ be a hyperbolic orbifold, and let $\lambda^{(0)}_1(X)$ denote the first non-zero eigenvalue of its Laplacian. Let $E_{\mathtt{spin}}\subset\mathbb{R}_{>0}$ be the set of these eigenvalues as $X$ ranges over all compact hyperbolic spin orbifolds. For an orbifold of signature $[0;k_1,k_2,k_3]$, write $\lambda_1^{[k_1,k_2,k_3]}$ for $\lambda^{(0)}_1([0;k_1,k_2,k_3])$. Spectral-gap conjecture. The set $E_{\mathtt{spin}}$ is

$$\left(0,\lambda_1^{[3,3,9]}\right]\cup\left\{\lambda_1^{[3,3,5]},\lambda_1^{[3,5,5]},\lambda_1^{[3,3,7]}\right\}.$$

The conjecture gives a proposed complete description of the possible first non-zero Laplacian eigenvalues for compact hyperbolic spin orbifolds, based on numerical data computed using the STF; its resolution is not indicated in the supplied text.