Non-Boltzmann limiting dynamics for periodic quantum transport

Let $L(t)$ denote the limiting family of linear operators arising in the scaling limit where the quantum wavelength and scattering radius satisfy $h=r\to0$, with observables evolved on the time scale $t r^{1-d}$. For suitable Weyl symbols $a,b\in\mathcal S(\mathbb R^d\times\mathbb R^d)$, write $A=\operatorname{Op}_{r,h}(a)$ and $B=\operatorname{Op}_{r,h}(b)$. Non-Boltzmann limit conjecture. There exists a family of linear operators

$$L(t):\operatorname{L}^1(\mathbb R^d\times\mathbb R^d)\to\operatorname{L}^1(\mathbb R^d\times\mathbb R^d)$$

such that, for all such $a,b$ and $t>0$,

$$\lim_{h=r\to0}\langle A(t r^{1-d}),B\rangle_{\operatorname{HS}}=\langle L(t)a,b\rangle,$$

while $L(t)a(\boldsymbol{x},\boldsymbol{y})$ is in general not a solution of the linear Boltzmann equation. This formulates the expected higher-order limiting dynamics in the periodic setting; the supplied text does not establish whether the assertion is proved or remains open.