Cilleruelo's minimum-degree conjecture for infinite $C_4$-free graphs

Let $\mathcal{G}$ be a countable infinite graph with no cycle of length four, and let $\mathcal{G}_n$ be the subgraph induced by its first $n$ vertices. Write $\delta(\mathcal{G}_n)$ for its minimum degree. Cilleruelo's conjecture.

$$\liminf_{n\to\infty}\frac{\delta(\mathcal{G}_n)}{\sqrt n}=0.$$

Finite $C_4$-free graphs can have minimum degree asymptotic to $\sqrt n$, but the author explains that algebraic constructions are too rigid to extend to infinite graphs. The conjecture asserts that every countable infinite $C_4$-free graph must have substantially smaller minimum degree along an infinite subsequence.