Edge-transitive graph conjecture for weak concentration and unique giant components

Let $(\mathbf V_n)$ be a sequence of edge-transitive graphs with $|\mathbf V_n|\to\infty$, equipped with edge-weights, and suppose the edge-weight can be rescaled so that the background assumption (hypothesis (2)) holds. The weak concentration and unique giant component properties concern the percolation component-size process, with the latter meaning

$$\sup_t \frac{C_n^{[2]}(t)}{|\mathbf V_n|}\to_p 0\quad\text{as }n\to\infty,$$

where $C_n^{[2]}(t)$ is the size of the second-largest component at time $t$.

Edge-transitive graph conjecture. One can always rescale the edge-weight so that the background assumption holds; after such rescaling, both the weak concentration and unique giant component properties hold.

Edge-transitivity makes all edge-weights equal, reducing the question to bond percolation on unweighted graphs with symmetry. The conjecture is motivated by results for random regular graphs, although their local weak limit is the regular tree, which itself does not have the unique giant component property. The source does not indicate a resolution.