The sharp logarithmic estimate conjecture for the constant $C_1$

Let $\mathcal X$ be either the space $B^0_{\infty,1}$ with a Lipschitz measure-preserving flow, or the space $L^p\cap\operatorname{LBMO}$ with the flow considered in the cited results. Suppose a logarithmic estimate has the form

\|f\circ\psi\|_{\mathcal X}\leq \left[C_1+C_2\ln\hspace{-3.8mm}{}^+\hspace{-3.8mm}{}^+\hspace{-3.8mm}{}^+\hspace{-3.8mm}{}^+\hspace{-3.8mm}{}^+\hspace{-3.8mm}{}^+\hspace{-3.8mm}{}^+\hspace{-3.8mm}{}^+^+(\|\psi\|_{C(\psi)})\right]\|f\|_{\mathcal X}.

Constant-$C_1$ conjecture. In both cases considered in the cited works, the constant $C_1$ can be taken equal to $1$.

The conjecture concerns the sharp value of the universal additive constant in logarithmic composition estimates arising in transport equations; the source does not state whether it has been resolved.