The converse capacity conjecture for quasi-Bernoulli tree-indexed processes

Let $(\Omega,S,B,\mu)$ be a tree-indexed process. Define

$$f(n)=\mu^n\bigl(\pi_n(B)\bigr)^{-1}$$

so that $f(n)$ is the probability that a sequence of $n$ IID picks from $\mu$ extends to some sequence in $B$. The converse capacity conjecture. If $P(\Gamma;B)>0$, then

$$\overline{\operatorname{cap}}_f(\Gamma)>0.$$

If true, this would make the capacity criterion necessary and sufficient in the quasi-Bernoulli case, complementing the Basic Theorem; the conjecture is identified in the source as the most important open problem for tree-indexed processes.