The one-third time-decay conjecture for the periodically forced Dirac propagator

Consider the periodically forced Dirac equation with mass $m=1$, and let $M^n$ denote its $n$-period propagator. For $\beta>0$, write $\langle\partial_x\rangle^\beta=(1-\partial_x^2)^{\beta/2}$. One-third time-decay conjecture. For every $\varepsilon>0$ there exists $C_{\varepsilon}>0$ such that

$$\left\|M^n\langle\partial_x\rangle^{-(2/3+\varepsilon)}\right\|_{L^1\to L^{\infty}}\leq C_{\varepsilon}n^{-1/3}.$$

The conjecture is motivated by numerically observed inflection points of the dispersion relation at arbitrarily large Fourier momenta, where the third derivative becomes small. Van der Corput estimates suggest the $n^{-1/3}$ decay, while the additional $\varepsilon$ of smoothing is attributed to the dyadic partition argument; the supplied source does not state whether this conjecture has been resolved.