Stable deformability conjecture for invariant distributions of geometrically finite groups

Let $\Gamma$ be a geometrically finite group, let $\sigma$ be a representation parameter, let $\lambda\in{\mathfrak a}_{\mathbb C}^*$, and let $\varphi$ be the auxiliary parameter appearing in the distribution space. Write ${\rm Fam}^{st}_\Gamma(\sigma_\lambda,\varphi)$ for the subspace of stably deformable $\Gamma$-invariant distributions in ${}^\Gamma C^{-\infty}(\partial X,V(\sigma_\lambda,\varphi))$. Stable deformability conjecture. If $\Gamma$ is geometrically finite, then

$${\rm Fam}^{st}_\Gamma(\sigma_\lambda,\varphi)={}^\Gamma C^{-\infty}(\partial X,V(\sigma_\lambda,\varphi))$$

for all $\sigma$, $\lambda$ and $\varphi$. This would extend the known result for convex cocompact groups to all geometrically finite groups; the source does not give a proof or resolution of the assertion.