Lindenstrauss–Tsukamoto embedding conjecture for topological dynamical systems

Let $(X,T)$ be a topological dynamical system, let $d$ be a positive integer, let $\operatorname{mdim}(X,T)$ denote its mean dimension, let $\operatorname{dim}$ denote topological dimension, and let $\sigma$ be the shift on $(([0,1]^d)^{\mathbb{Z}})$. Lindenstrauss–Tsukamoto conjecture. If

$$\operatorname{mdim}(X,T)<\frac{d}{2}$$

and

$$\frac{\operatorname{dim}(\{x:T^n x=x\})}{n}<\frac{d}{2}\quad\text{for all }n\geq 1,$$

then there is an embedding from $(X,T)$ into $(([0,1]^d)^{\mathbb{Z}},\sigma)$. This conjecture holds generically, but it remains widely open in general.