Log-Lipschitz Mané projector conjecture for dissipative PDE attractors

Let $H$ be the phase space, let $H^2$ denote the more regular space, let $F$ be the non-linearity satisfying the reasonable assumptions considered in the paper, and let $\mathcal A$ be the attractor. A Mané projector is a linear map $P\in\mathcal L(H,H)$ whose restriction to $\mathcal A$ is injective, with inverse defined on $\overline{\mathcal A}$. Log-Lipschitz Mané projector conjecture. The attractor $\mathcal A$ possesses at least one Mané projector $P\in\mathcal L(H,H)$ such that $P^{-1}:\overline{\mathcal A}\to\mathcal A\subset H^2$ satisfies, for some positive constants $C$ and $K$,

$$\|P^{-1}v_1-P^{-1}v_2\|_{H^2}\le C\|v_1-v_2\|_H\left(\log\frac{2K}{\|v_1-v_2\|_H}\right)^\alpha,\qquad v_1,v_2\in\mathcal A,$$

with exponent $\alpha\le 1$. The question arises from the existence of bi-Lipschitz Mané projections under stronger estimates and the weaker log-Lipschitz estimate established earlier in the paper. The conjecture is refuted: the paper states that it was shown in the cited work [EKZ] to be wrong, with a corresponding counterexample discussed in the following section.