Percolation transition for height-three vertices

For $n\in\mathbb{N}$, let $\mu_n^{(p)}$ be the stationary distribution with $p$-topplings on the square box $[[1,n]]^2$, with particles lost outside the box. For a sandpile configuration $\eta$, define the height-three vertex set by

$$T_n(\eta):=\{v\in[[1,n]]^2:\eta(v)=3\}.$$

Height-three percolation conjecture. There exists $p_c\in(0,1)$ such that, for all $p<p_c$,

$$\mu_n^{(p)}(\text{There exists a path in }T_n(\eta)\text{ that connects the left and right boundary})\xrightarrow{n\to\infty}1,$$

and, for all $p>p_c$,

$$\mu_n^{(p)}(\text{There exists a path in }T_n(\eta)\text{ that connects the left and right boundary})\xrightarrow{n\to\infty}0.$$

This conjectures a phase transition in horizontal crossing by height-three vertices as the toppling parameter varies; simulations suggest a transition near $p\approx0.64$, but no resolution is supplied.