Ising-eroding conjecture for critical families

Let $\mathcal U$ be a finite family defining two-dimensional $\mathcal U$-Ising dynamics. A finite set $\mathcal T$ of stable directions determines a $(\mathcal T,L)$-droplet when the intersection of the corresponding translated discrete half-planes is nonempty, finite, and has diameter at most $L$. The family is Ising-eroding if there exist $c>1$ and a finite $\mathcal T$ such that every $(\mathcal T,L)$-droplet $D$ satisfies

$$\partial_p\bigl(T^{\operatorname{Is}}(D)>L^c\bigr)\leqslant e^{-L}$$

for all sufficiently large $L$, where $T^{\operatorname{Is}}(D)$ is the time required for the droplet, initially all $-$ with all exterior sites frozen at $+$, to become entirely $+$. Ising-eroding conjecture. Every critical family is Ising-eroding.

The conjecture is motivated by the known polynomial-time erosion estimate for the nearest-neighbour family $\mathcal N_2^2$ and by numerical simulations. The source describes it as difficult to prove and does not establish it.