Smoothness conjecture for the exponential attractor under finite dissipation

Let $f=f_B$, and let $k$ be sufficiently large depending on the internal parameters $Omega$, $L$, $b$, and $p$. There is a proper fractal exponential attractor

$$\mathcal A_{\text{exp}}\subset Y$$

with finite fractal dimension in $Y$ for the dynamics $(S_t,Y)$. If the dissipation integral is finite, namely

$$\int_T^\infty\lVert u_t\rVert^2\,d\tau<\infty,$$

then the fractal exponential attractor satisfies

$$\mathcal A_{\text{exp}}\subset W,$$

and is bounded in the topology of $W$. Smoothness conjecture. Under the stated assumptions, finite dissipation implies that the fractal exponential attractor is bounded in the stronger topology $W$. The source presents this as a conjectural consequence for the reduced plate model; no resolution is supplied.