Slow-dispersion conjecture for Floquet-Dirac Hamiltonians

Let $\nu(t)$ be a choice of potential in the Floquet-Dirac Hamiltonian, let $\alpha(t,x)$ denote the corresponding solution, and let $\langle i\partial_x\rangle^r$ be the indicated regularity operator for $r\geq 0$. For a decay exponent $\sigma>0$, consider an estimate of the form

$$\|\alpha(t,\cdot)\|_{L^\infty_x}\leq \frac{c}{t^{\sigma}}\|\langle i\partial_x\rangle^r\alpha(0,\cdot)\|_{L^1_x}.$$

Slow-dispersion conjecture. For every $\varepsilon>0$, there exists a choice of $\nu(t)$ such that the $L^1\to L^\infty$ decay rate is no faster than $t^{-\varepsilon}$; more precisely, for every $r\geq 0$, the displayed estimate requires

$$0<\sigma\leq\varepsilon.$$

The conjecture asserts that suitable Floquet potentials can make dispersive decay arbitrarily slow, extending the paper's construction, which uses four parameters to eliminate the first nine derivatives of the dispersion relation and make the tenth derivative dominant. Its status is not established by the supplied context.