co.combinatorics – Can cycle homomorphisms dominate cycle subgraphs in dense enough graphs?

Let $k$ be a fixed positive integer and let $G$ be an $n$-node graph with average degree $d ge C_k n^{1/k}$. It is known that (if $C_k$ is large enough) there are $Omega_k(d^{2k})$ homomorphisms from a $2k$-cycle into $G$, and even that there are $Omega_k(d^{2k})$ instances of $C_{2k}$ as a subgraph, and that both of these lower bounds are best possible save the hidden constant.

Thus, on graphs where these lower bounds are both tight, a constant fraction of the homomorphisms are injections, i.e. correspond to subgraphs. My question is whether this is true of all graphs: if $G$ satisfies the above density condition, does this necessarily mean that a constant fraction $c_k > 0$ of its cycle homomorphisms are injective? I would guess that the answer is yes, but I haven’t been able to find this in prior work.