Uniform observability conjecture for Schrödinger equations on hyperbolic coverings

Let $M$ be a compact hyperbolic surface, let $X$ be a $\Gamma$-covering of $M$, and let $S\subset X$ be a nonempty, $\Gamma$-periodic, open subset. Write $\pi:X\to M$ for the covering map and set $\Omega=\pi(S)$. For $u\in L^2(X)$, the observability inequality is the estimate

$$\|u\|_{L^2(X)}^2\leq C\int_0^T\|e^{it\Delta_g}u\|_{L^2(\pi^{-1}(\Omega))}^2\,dt.$$

Uniform observability conjecture. For every $T>0$, there exists a constant $C=C(X,S,T)>0$ such that the observability inequality holds. Moreover, the constant can be chosen to depend only on $M$, $\Omega$, and $T$.

The conjecture would extend the observability theorem beyond coverings by type I groups to arbitrary $\Gamma$-coverings of compact hyperbolic surfaces. The source gives no resolution, so its status remains open.