Mean-dimension embedding conjecture for real flows

Let $(X,\mathbb{R})$ be a flow and let $a>0$ be a real number. For $r\in\mathbb{R}$, write

$$P_r(X,\mathbb{R})=\{x\in X:rx=x\}.$$

Mean-dimension embedding conjecture. If

$$\operatorname{mdim}(X,\mathbb{R})<a$$

and, for every $r\in\mathbb{R}$,

$$\dim P_r(X,\mathbb{R})<\lfloor ar\rfloor+\frac{1}{2},$$

then $(X,\mathbb{R})$ can be embedded in the flow $(B_1(V^{\mathbb{R}}[-a,a]),\mathbb{R})$. This is the proposed continuous-time analogue of the Lindenstrauss–Tsukamoto embedding conjecture, with the band-limited flow as the target; the supplied source does not establish whether the conjecture is resolved.