Higher-dimensional universality conjecture for critical bootstrap percolation families

Let $\mathcal U$ be a critical $d$-dimensional family, meaning that its stable directions satisfy the criticality condition, and let $L_c(\mathcal U,p)$ be the critical length at initial density $p$. For an integer $r$, write $\log_{(r-1)}$ for the $(r-1)$-fold iterated logarithm. Higher-dimensional universality conjecture. There exists $r\in\{2,\dots,d\}$ such that, as $p\rightarrow 0$,

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

This is proposed as a weaker higher-dimensional analogue of the two-dimensional universality theorem, which gives a precise alternative between single- and doubly-exponential critical lengths. Establishing such a universality result for all critical families in dimensions at least three remains open.