Time-independent closure conjecture for Schrödinger-wave observability

For each $d\geq 1$, let $Y_0$ be the closure space for Schrödinger waves introduced in the paper, and let $\tilde{Y}_0\subset Y_0$ be its time-independent subspace. Time-independent closure conjecture. Every bounded time-independent function on $\mathbb{T}^d$ belongs to $\tilde{Y}_0$:

$$\tilde{Y}_0\supset L^\infty_x.$$

The conjecture is presented as a weaker counterpart to the closure conjecture for all bounded space-time functions. Its role is to provide approximation of bounded spatial observables in the norm relevant to Schrödinger eigenfunctions; no resolution is supplied in the context.