The third-derivative conjecture for the first band function at its minimum

Let $\mu_{1}$ be the first band function, let $\xi_{0}$ denote its minimum point, and write $\mu_{1}^{(3)}(\xi_{0})$ for its third derivative there. Third-derivative conjecture.

$$\mu_{1}^{(3)}(\xi_{0})<0.$$

This sign condition is supported by numerical simulations and is assumed in part of the subsequent analysis; its general validity is not established in the supplied text.