Universal inverse-degree upper bound for critical percolation

Let $G$ be an infinite, connected, transitive, simple graph of vertex degree $d$ that is not one-dimensional, and let $p_c(G)$ be its Bernoulli bond percolation critical probability. Universal inverse-degree conjecture. There exists a universal constant $C$ such that

$$p_c(G)\leq\frac{C}{d}.$$

This conjecture would strengthen known uniform gaps below $1$ for Cayley graphs and would imply, together with locality, that $p_c$ attains a global maximum on the space of non-one-dimensional transitive graphs. The source leaves the conjecture open.