Develin–Hartke lower-bound conjecture for firefighting on integer lattices

Let $d$ be the growth degree of the integer lattice $\mathbb{Z}^d$, and let $f\colon\mathbb{N}\to\mathbb{N}$ satisfy

$$f(n)=o(n^{d-2}).$$

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

This conjecture proposes a converse to the known $O(n^{d-2})$ containment upper bound for $\mathbb{Z}^d$, relating the minimum firefighter deployment rate to the lattice's polynomial growth. The supplied context does not state whether the conjecture has been resolved.