Sub-polynomial boundary-growth conjecture for fire containment in d-dimensional grids

Let $\mathbb{L}^{d}$ be the infinite $d$-dimensional square grid, whose vertices are the points of $\mathbb{R}^{d}$ with integer coordinates, with adjacency defined by Euclidean distance $1$. Let $f(t)$ be the number of firefighters deployed at time $t$, where $f$ is a function on $\mathbb{N}$. Sub-polynomial boundary-growth conjecture. If

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

as $t$ tends to infinity, then there exists an outbreak on $\mathbb{L}^{d}$ that cannot be contained by deploying $f(t)$ firefighters at time $t$. This is motivated by the fact that in dimensions $d\geq 3$, outbreak boundaries grow polynomially of degree $d-2$, but the source presents the claim as an ambitious conjecture and gives no resolution.