The density-of-the-closure conjecture for subcritical $d53c$-bootstrap percolation

Let $A$ be the initially occupied random set in a subcritical $d53c$-bootstrap percolation model on $d539^2$, let $[A]$ be its closure, and define

$$D(n)=\frac{\left|[-n,n]^2\cap[A]\right|}{\left|[-n,n]^2\right|}.$$

Density-of-the-closure conjecture. For every $p\in[0,1]$, there exists a constant $\delta(p)$ such that $D(n)$ converges in probability to $\delta(p)$ as $n\rightarrow\infty$.

The conjecture asserts that sites in the closure should be reasonably well scattered. If true, questions about continuity of $\delta(p)$ at $p=p_c$ and the behaviour of $\delta(p)-p$ as $p\rightarrow0$ remain open.