The scout-number conjecture for effective exploration of integer grids

A scout process consists of $c$ scouts moving on $\mathbb{Z}^d$ according to a scout protocol $\mathcal{P}=\langle c,\mathcal{S},x_0,\mathbf{q}_0,\Pi\rangle$. The hitting time of $x\in\mathbb{Z}^d$ is $\inf\{n\geq 0:\exists i\text{ such that }X_n^i=x\}$, and the protocol is effective when every grid point has finite mean hitting time. It is known that $d+1$ scouts suffice for every $d\geq 1$.

The scout-number conjecture. For $d\geq 3$, any effective scout protocol on $\mathbb{Z}^d$ requires at least $d+1$ scouts.

The claim extends the paper's theorem for $d\in\{1,2\}$ and, together with the known sufficiency of $d+1$ scouts, would identify the exact minimum number of scouts needed for effective exploration of $\mathbb{Z}^d$.