Metastability scaling conjecture for critical two-dimensional update families

Let ${\mathcal U}$ be a critical two-dimensional update family. Its difficulty is

$$\alpha=\min_{u\in S^1}\max_{v\in S^1:\langle u,v\rangle>0}\alpha(v),$$

where a critical update family is balanced if there are no two opposite directions $v,-v\in S^1$ with $\min(\alpha(v),\alpha(-v))>\alpha$, and unbalanced otherwise. Let $\tau$ denote the percolation time and let $\gamma=0$ in the balanced case and $\gamma=2$ in the unbalanced case. Metastability scaling conjecture. There exists a constant $\lambda=\lambda({\mathcal U})$ such that, for every $\varepsilon>0$,

$$\lim_{p\to 0}{{\mathbb P}} _p\left(\left|\frac{p^\alpha\log\tau}{(\log(1/p))^\gamma}-\lambda\right|>\varepsilon\right)=0.$$

This conjecture proposes a common sharp metastability scale for all critical two-dimensional update families, with a logarithmic correction distinguishing balanced from unbalanced models. The supplied passage presents it as a future generalisation of the proved result for symmetric isotropic threshold rules; its resolution for general critical models is not stated.