Three-dimensional critical-family universality conjecture for bootstrap percolation

Let $\mathcal U$ be a critical three-dimensional family, and let $L_c(\mathcal U,p)$ denote its critical length at probability $p$. As $p\rightarrow 0$, either

$$\log L_c(\mathcal U,p)=p^{-\Theta(1)},$$

or

$$\log\log L_c(\mathcal U,p)=p^{-\Theta(1)}.$$

Critical-family universality conjecture. As $p\rightarrow 0$, either

$$\log L_c(\mathcal U,p)=p^{-\Theta(1)},$$

or

$$\log\log L_c(\mathcal U,p)=p^{-\Theta(1)}.$$

This is a weaker universality prediction for critical bootstrap-percolation families in dimension three. The corresponding universality result is known in dimension two, while proving a result of this kind in three or higher dimensions is described as an open problem.