Uniqueness conjecture for the joint fixed point of the consistency equation

Let $(U,X)$ be a random pair satisfying the consistency equation

$$(U,X)\stackrel{d}{=}\Bigl( 1-\sum_{k=1}^{R}\frac{2\beta\zeta_k\xi_kX_k U_k}{1+2\beta\xi_k^2 X_k},\Bigl(1+\sum_{k=1}^{R}\frac{2\beta\zeta_k^2}{1+2\beta \xi_k^2X_{k}}\Bigr)^{-1}\Bigr),$$

where $(U_k,X_k)_{k\geq 1}$ are independent copies of $(U,X)$, $(\xi_k)_{k\geq 1}$ and $(\zeta_k)_{k\geq 1}$ are independent copies of $\mathcal{D}$, and $R$ is Poisson with mean $2\alpha$, with all these variables independent. Uniqueness conjecture. The consistency equation has a unique fixed point in the law of $(U,X)$. The uniqueness of the second coordinate has already been established, while uniqueness of the joint fixed point remains to be shown.