Multiplicity one conjecture for restriction between reductive groups

Let ${\mathbf{G}}$ be a quasi-split reductive algebraic group over a local field $k$. Let $\widetilde{{\mathbf{G}}}$ be a reductive algebraic group containing ${\mathbf{G}}$ such that the derived groups of ${\mathbf{G}}$ and $\widetilde{{\mathbf{G}}}$ are the same, and such that $\widetilde{{\mathbf{G}}}/{\mathbf{G}}$ is connected.

Multiplicity one conjecture. Multiplicity one holds for restriction of irreducible admissible representations of $\widetilde{{\mathbf{G}}}(k)$ to ${\mathbf{G}}(k)$.

This conjecture seeks a general multiplicity one theorem extending the paper's results for restriction from $\operatorname{GU}(V)$ to $\operatorname{U}(V)$ and from $\operatorname{GL}(n)$ to $\operatorname{SL}(n)$. The authors suggest that it might be proved by analyzing invariant distributions, in analogy with the uniqueness of Whittaker models for quasi-split groups.