Invariant bisolution basis conjecture for the universal cover of anti-de Sitter space

Let $W_0$ denote the universal cover of anti-de Sitter space, and let $Z$ be the invariant anti-de Sitter quantity appearing in the Gegenbauer kernels. For the relevant parameter $\nu$, consider the four bisolutions proportional to

$$\mathbf{Z}_{\frac{d}{2}-1,\nu}\big(-Z\pm\mathrm{i}0\operatorname{sgn}(\tau)\big)$$

and

$$\mathbf{Z}_{\frac{d}{2}-1,-\nu}\big(-Z\pm\mathrm{i}0\operatorname{sgn}(\tau)\big),$$

where the two signs are chosen independently as in the displayed family in the source. Invariant bisolution basis conjecture. On $W_0$, these four functions form a basis of the bisolutions of the Klein–Gordon equation invariant under the restricted anti-de Sitter group. The claim concerns the classification of invariant Klein–Gordon bisolutions on the universal cover, where the obstruction at the antipodal diagonal present on proper anti-de Sitter space disappears; the supplied text gives no resolution, so the conjecture remains open.