Asymptotic tightness conjecture for limited-visibility cops on Hamming graphs

Let $H(d,n)$ be the Hamming graph and let $c'_{d-1}(H(d,n))$ denote the minimum number of cops needed when each cop has visibility limited to $(d-1)$-slices. For any fixed positive integer $d$, the limited-visibility cop number satisfies

$$c'_{d-1}(H(d,n))=(1+o(1))\frac{n}{d+1}.$$

Asymptotic tightness conjecture. For any fixed positive integer $d$, we have

$$c'_{d-1}(H(d,n))=(1+o(1))\frac{n}{d+1}.$$

The paper establishes lower and upper bounds of order $n$ for this cop number and argues that the lower bound should be asymptotically tight; the conjecture asserts that the lower bound has the correct asymptotic constant for every fixed $d$.