Giant-component conjecture for growing-girth cubic graphs

Let $G_n$ be a sequence of finite 3-regular graphs with growing girth, converging locally to the 3-regular tree $T_3$, and assume that the thresholds $p^{G_n}$ remain bounded away from $1$. For bond percolation, let $\mathbf{P}_p$ denote the percolation probability on $G_n$. Cubic giant-component conjecture. For every $p>1/2$, there is $C_p>0$ such that

$$\mathbf{P}_p\bigl(\text{there is a unique connected component of size }C_p|G_n|\bigr)\longrightarrow 1$$

as $n\to\infty$. Since $p_c(T_3)=1/2$, this predicts a linear unique giant above the limiting tree threshold; the paper gives a random-matching construction as supporting evidence, but the general claim is open.