The L1 convergence conjecture for destinations of flip processes

Let $\mathcal{R}$ be a flip process, let $U$ be a graphon, and let $W$ be a destination of $U$, meaning that $\Phi^{t}U \xrightarrow{\square} W$ as $t\to\infty$. Destination convergence conjecture. The trajectory $(\Phi^{t}U)_{t\to\infty}$ converges to $W$ also in $L^1$. Destinations are defined initially as cut norm limits; this conjecture asks for the stronger $L^1$-norm convergence, and the surrounding discussion suggests that even $L^\infty$-norm convergence might hold.