Zero–infinity conjecture for packing measures of non-almost-deterministic random code-tree attractors

Let $bbL$ be a family of iterated function systems satisfying the UOSC and Condition~. Let $bbP$ be a random code-tree measure, and assume that $F_\tau$ is not almost deterministic. Let $h(t)$ be any gauge function. Zero–infinity conjecture.

$$bbP\left\{\tau\in\mathcal{T}\mid \mathscr{P}^h(F_\tau)\in\{0,\infty\}\right\}=1.$$

In particular, no gauge function gives positive and finite packing measure almost surely. The conjecture asserts that non-almost-deterministic random code-tree attractors cannot have a gauge with positive and finite packing measure almost surely; establishing this would complete the expected zero–infinity alternative for their packing measures.