Hammel–Yorke–Grebogi finite Hölder shadowing conjecture

Let $f:\mathbb{R}^2\to\mathbb{R}^2$ be a dissipative map. For $\theta\in(0,1)$ and $\omega\geq 0$, say that $f$ has the finite Hölder shadowing property $\operatorname{FinHolSh}(\theta,\omega)$ if there exist constants $d_0,L,C>0$ such that, for every $d<d_0$, every $d$-pseudotrajectory $\{y_k\}_{k\in[0,Cd^{-\omega}]}$ is shadowed by an exact trajectory $\{x_k\}_{k\in[0,Cd^{-\omega}]}$ satisfying

$$\operatorname{dist}(x_k,y_k)<Ld^\theta,\qquad k\in[0,Cd^{-\omega}].$$

Hammel–Yorke–Grebogi conjecture. A typical dissipative map $f:\mathbb{R}^2\to\mathbb{R}^2$ satisfies

$$\operatorname{FinHolSh}(1/2,1/2).$$

This conjecture concerns finite-time Hölder shadowing for typical dissipative planar dynamics and was motivated by numerical experiments; the source does not report a resolution.