Weak-strong uniqueness for rough Landau initial data

Let $f_{\rm in}$ be the initial datum in the setting of the paper's Landau weak-strong uniqueness theorem, and let $L^{\infty,k}$ denote the weighted bounded space used there. The notation $\log(1/C_v)^{-\theta}$ denotes the stated logarithmic velocity regularity class. Weak-strong uniqueness conjecture. In the setting of the cited Landau theorem, even if one assumes only

$$f_{\rm in}\in \log(1/C_v)^{-\theta}\cap L^{\infty,k},$$

so that the Hölder regularity in $x$ is dropped, the same weak-strong uniqueness conclusion holds provided $\theta>2$. The conjecture concerns uniqueness for rough solutions and is motivated by the preceding conjectured kinetic Schauder estimate; the supplied status evidence says that uniqueness of these rough solutions remains open.