Sharp asymptotics and monotonicity conjecture for three-dimensional anisotropic bootstrap percolation

Let $p$ be the initial occupation probability, and let $L_{a,b,c}^{th}(p)$ denote the threshold length for the three-dimensional $(a,b,c)$ model. Let $f_{a,b}(1/p)$ be the function specified in the paper's main theorem, and let $C_{a,b,c}$ be a model-dependent constant.

Three-dimensional threshold conjecture. There is a constant $C_{a,b,c}$ such that

$$\ln\ln L_{a,b,c}^{th}(p)=C_{a,b,c}f_{a,b}(1/p)+o\bigl(f_{a,b}(1/p)\bigr).$$

If $a'\geq a$, $b'>b$, and $c'\geq c$, then

$$\ln\ln L_{a',b',c'}^{th}(p)\geq\ln\ln L_{a,b,c}^{th}(p).$$

The paper's main result determines the order of $\ln\ln L_{a,b,c}^{th}(p)$ up to multiplicative constants, but does not establish these sharp asymptotics or the asserted monotonicity in general.