Conjecture on the cop number of fast robbers on grids

For $R\in\mathbb{N}$, let $f_R(n)$ be the minimum number of cops needed to catch a robber of speed $R$ on an $n\times n$ grid, where the cops move at unit speed. Here “sufficiently large” refers to the robber speed parameter $R$.

Fast-robber grid conjecture. For all sufficiently large $R\in\mathbb{N}$,

$$f_R(n)=n^{1-o(1)}\quad\text{as }n\to\infty.$$

The conjecture says that, for sufficiently large fixed robber speeds, the trivial upper bound $f_R(n)\le n$ is asymptotically close to the truth. The paper proves a much weaker exponential lower bound for one sufficiently large speed and leaves this near-linear behaviour conjectural.