Invariant bisolution basis conjecture for de Sitter space

Let $d$ be the spacetime dimension, let $u\in\mathbb{C}$ satisfy

$$\frac{d-1}{2}\pm\mathrm{i}\nu \notin\\{0,-1,-2,\dots\\},$$

and let $G_0^{\mathrm{sym}}$, $G_0^{\mathrm{sym},A}$, $G^{\mathrm{PJ}}$, and $G^{\mathrm{PJ},A}$ be the invariant bisolutions defined above, where the superscript $A$ denotes replacement of one argument by its antipode. Invariant bisolution basis conjecture. The pair $\\{G_0^{\mathrm{sym}},G_0^{\mathrm{sym},A}\\}$ is a basis of the space of fully de Sitter invariant bisolutions, and the four functions $\\{G_0^{\mathrm{sym}},G_0^{\mathrm{sym},A},G^{\mathrm{PJ}},G^{\mathrm{PJ},A}\\}$ are a basis of the space of bisolutions invariant under the restricted de Sitter group. This would classify the invariant bisolutions relevant to the propagators considered in the paper; the supplied text presents the assertion as an expectation and gives no resolution, so its status remains open.