Develin–Hartke conjecture on subcritical containment in square grids

Let $d$ be a positive integer, let $\mathbb{L}^d$ denote the $d$-dimensional square grid, and let $f\colon\mathbb{N}\to\mathbb{R}$ satisfy

$$\frac{f(t)}{t^{d-2}}\longrightarrow 0$$

as $t$ tends to infinity. An outbreak is a fire spreading through the vertices of $\mathbb{L}^d$, and deploying $f(t)$ firefighters at time $t$ means that at most that many vertices can be protected at that time.

Develin–Hartke conjecture. There exists an outbreak on $\mathbb{L}^d$ that cannot be contained by deploying $f(t)$ firefighters at time $t$.

The claim predicts that any deployment rate asymptotically smaller than $t^{d-2}$ fails to contain some outbreak. The source notes a weaker version in which $f(t)$ is required to be a polynomial, and presents a corollary proving the polynomial case.