Higher-dimensional Borg uniqueness conjecture for periodic Schrödinger operators

Let $H=-\Delta+V(x)$ be a periodic Schrödinger operator, with dispersion relation consisting of the pairs $(k,\lambda)$ for which $H(k)$ has a non-zero solution at energy $\lambda$. An entire dispersion branch is an entire function $f(k)$ whose graph $\lambda=f(k)$ lies in the dispersion relation.

Higher-dimensional Borg conjecture. The following claims are equivalent:

1. The potential $V$ is constant.
2. There exists an entire function whose graph $\lambda=f(k)$ belongs to the dispersion relation.

This is proposed as the higher-dimensional analogue of Borg's uniqueness theorem for the one-dimensional Hill operator. The source explicitly says that the equivalence probably still holds and gives no resolution; the two-dimensional version is not the statement extracted here.