Generic rigidity conjecture for periodic potentials

Let $m$ denote the number of Fourier modes in the periodic potentials, let $n$ be the ambient dimension, and let $V_1,V_2$ be smooth periodic functions. The rigidity result referred to as Theorem 2d concerns the implication identifying potentials from equality of their effective Hamiltonians, with the stated translation-and-rational-scaling form in the theorem. Generic rigidity conjecture. (1) Theorem 2d should hold when $m\geq 3$ and $n=2$. (2) If $n\geq 3$, then the result of Theorem 2d should be valid provided that $V_1$ and $V_2$ belong to a dense open set of smooth periodic functions. This extends the established rigidity results beyond the cases $m\leq 3$ and predicts rigidity for generic smooth periodic potentials in higher dimensions.