Sharp asymptotics for unbalanced alpha-unrooted critical update families

Let ${\mathcal U}$ be an $\alpha$-unrooted and unbalanced critical two-dimensional update family with difficulty $\alpha$. Let $q$ denote the vacancy density and let $\tau_0$ be the infection time at the origin. Sharp-asymptotics conjecture. As $q\rightarrow0$,

$${\mathbb E}_\mu(\tau_0)=\exp\!\left(\Theta\!\left(q^{-\alpha}(\log q^{-1})^2\right)\right).$$

This would identify the precise leading-order exponent in the unbalanced case. The paper notes that its upper and lower bounds currently differ by only one factor of $\log(1/q)$ in the exponent.