Multi-layer analogue of Meyniel's conjecture

Let $\tau\geqslant 1$ be a fixed integer, and let $G$ be a connected graph on $n$ vertices. Write $\mathsf{emc}_{\tau}(G)$ for the extremal multi-layer cop number over multi-layer graphs on $G$ with $\tau$ connected layers.

Multi-layer analogue of Meyniel's conjecture. For every fixed $\tau\geqslant 1$ and every connected $n$-vertex graph $G$, one has

$$\mathsf{emc}_{\tau}(G)=o(n).$$

If layers may be disconnected, or if the number of layers grows arbitrarily with $n$, the source gives examples showing that no bound better than $\mathcal{O}(n)$ is possible. The conjecture asks whether a sublinear bound holds when the number of connected layers is fixed.