Finite determining set conjecture for the entire flow-plate dynamics

Let $(S_t,Y)$ denote the entire flow-plate dynamics, and let $k_*$ be the damping threshold from the convergence-to-equilibria theorem. Assume the hypotheses of that theorem, in particular $k>k_*$. A set $\mathbb L$ is called a finite determining set for $(S_t,Y)$ when its finitely many functionals determine the asymptotic behavior of the entire dynamics.

Finite determining set conjecture. There exists a set $\mathbb L$ which is a finite determining set for the entire dynamics $(S_t,Y)$.

The conjecture concerns the no-rotational-inertia case of the model and is motivated by determining-functional results for systems with rotational inertia, where positive damping suffices. It is presented as speculative and remains open in the source.