The maximum-degree bound for lazy cop number

Let $G$ be a graph on $n$ vertices, let $\Delta$ be its maximum degree, and let $c_L(G)$ denote its lazy cop number.

Maximum-degree conjecture. If

$$\Delta\geq n-k^2,$$

then

$$c_L(G)\leq k.$$

The statement is proposed as a natural generalization of the paper's proved bound for maximum degree at least $n-9$. The source suggests that it would follow inductively from the rook graph conjecture, but does not establish it.