Turcotte's cop-number conjecture for graphs with bounded independence number

For a graph $G$, let $\b1(G)$ denote its independence number, and let $c(G)$ denote the minimum number of cops needed to capture a robber on $G$.

Turcotte's conjecture. For any positive integer $p$ and any graph $G$ such that $\alpha(G)<p$, we have

$$c(G)\leq p-2.$$

This conjecture proposes a general upper bound on the cop number in terms of the independence number. It was raised by Turcotte and is presented here as an open conjecture; the supplied material gives no evidence of a resolution.