Gutman–Qiao–Tsukamoto embedding conjecture for band-limited flows

Let $(X,T)$ be a topological dynamical system, let $a$ be a positive real parameter, let $\operatorname{mdim}(X,T)$ denote its mean dimension, let $\operatorname{dim}$ denote topological dimension, and let $\mathcal{B}(a)$ be the flow space used in the paper. Gutman–Qiao–Tsukamoto conjecture. If

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

and

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

then there is an embedding from $(X,T)$ into $\mathcal{B}(a)$. This conjecture is motivated by the sampling-theoretic embedding of $(\mathcal{B}(a),\mathbb{Z})$ into the full shift when $a<d$; the paper records it as open.