Cilleruelo's lower-bound conjecture for infinite $C_4$-free graphs

Let $\epsilon>0$. A countable infinite graph with no cycle of length four is called $C_4$-free, and let $\mathcal{G}_n$ denote the subgraph induced by the first $n$ vertices. Write $\delta(\mathcal{G}_n)$ for its minimum degree. Cilleruelo's conjecture. For every $\epsilon>0$, there exists a countable infinite $C_4$-free graph $\mathcal{G}$ such that

$$\delta(\mathcal{G}_n)\gg n^{1/2-\epsilon}.$$

This conjecture proposes that, despite the obstruction posed by extending finite algebraic constructions to infinite graphs, infinite $C_4$-free graphs can still have minimum degree close to the finite-graph scale $\sqrt n$.