Resistance exponent conjecture for stable causal carpets

Let $bc$ be critical and satisfy the stable offspring condition for some $\alpha\in(1,2)$, and let the associated causal carpet be obtained by retaining only the extreme-most horizontal connections at each branch point. Let $\mathfrak{r}$ denote its resistance growth exponent.

Resistance exponent conjecture. The exponent $\mathfrak{r}$ exists and satisfies

$$0<\mathfrak{r}<1$$

almost surely. In particular, the causal carpet is recurrent almost surely.

This conjecture is motivated by the resistance bound obtained for causal carpets and, at $\alpha=3/2$, by the connection with subgraphs of the uniform infinite planar triangulation. Proving the conjecture would in particular yield a sublinear resistance upper bound for the uniform infinite planar triangulation; the statement remains open.