Uniform-integrability closure conjecture for Schrödinger-wave observability

For each $d\geq 1$, let $Y$ be the normed function space introduced in the paper for Schrödinger-wave observability, and let $Y_0$ denote the closure of continuous functions in $Y$. Uniform-integrability closure conjecture. Every bounded function on $\mathbb{T}^{1+d}$ belongs to $Y_0$:

$$Y_0\supset L^\infty_{t,x}.$$

Together with the paper's observability criterion, this would extend observability to all bounded space-time weights or observation functions covered by that criterion. The supplied context does not provide the definition of the $Y$-norm or indicate that the conjecture has been resolved.