Equality of critical parameters for loop and Bernoulli percolation at diverging degree

Let $(G_n)_n=(V_n,E_n)_n$ be a sequence of finite, connected, vertex-transitive graphs with vertex degree $\bDelta_n$ diverging as $n\to\infty$. For a fixed sequence and fixed $u$, define

$$t_p:=\inf\{t:\exists c>0:\lim_n\pi_{G_n,p=t/\Delta_n}(|C(v)|>c|V_n|)>0\},$$

and

$$t_\beta:=\inf\{t:\exists c>0:\lim_n\mathbb{P}_{G_n,\beta=t/\Delta_n,u}(|L(v)|>c|V_n|)>0\},$$

whenever the limits are well-defined, and set $p_c((G_n)_n):=t_p/\Delta_n$, $\beta_c((G_n)_n,u):=t_\beta/\Delta_n$, and $\beta_c^\text{per}((G_n)_n):=-\ln(1-p_c((G_n)_n))$. Critical-parameter equality conjecture. If $p_c((G_n)_n)$ and $\beta_c((G_n)_n,u)$ exist, then for every $u\in[0,1]$,

$$\beta_c((G_n)_n,u)=\beta_c^\text{per}((G_n)_n).$$

The conjecture proposes that, even when the vertex degree diverges, the loop-percolation critical parameter agrees with the corresponding Bernoulli-percolation parameter. The source gives no resolution, and the vertex-transitivity assumption is explicitly not expected to be necessary.