Positive-measure observability conjecture for Schrödinger eigenfunctions on tori

Let $d\geq 1$, let $\mathbb{T}^d$ be the $d$-dimensional torus, and let $V\in L^\infty_x(\mathbb{T}^d)$. Let $\mathcal{H}_V=-\Delta+V$, and let $f$ be any eigenfunction of $\mathcal{H}_V$. Eigenfunction observability conjecture. For every measurable $\omega\subset\mathbb{T}^d$ with positive Lebesgue measure, one has

$$\|f\|_{L^2_x}\lesssim_{d,V,\omega}\|f\|_{L^2_x(\omega)}.$$

This is the stationary analogue of the space-time observability conjecture and asserts quantitative unique continuation from every positive-measure set. The source presents it as a conjecture for bounded potentials in arbitrary dimension.