Conjecture on the cop number of graphs forbidding an induced cycle

For an integer $\ell\geq 3$, let $C_\ell$ be the cycle with $\ell$ vertices. A graph $G$ is $C_\ell$-free if it contains no induced subgraph isomorphic to $C_\ell$; write $c(G)$ for its cop number and $\alpha(G)$ for its independence number. Induced-cycle-free cop-number conjecture. For every integer $\ell\geq 3$, almost every $C_\ell$-free graph $G$ satisfies

$$c(G)<\alpha(G).$$

This extends the paper's motivation from the case of $C_4$-free graphs, for which the asserted inequality is proved for almost every graph, to all forbidden induced cycles. The source presents the statement as a conjecture and gives no resolution.