Conjecture on collapse and average logarithmic integer complexity

Let $\bar{f}_{\log}(n)$ denote the average logarithmic integer complexity, and let $\alpha_{ave}$ be the corresponding limiting average complexity. Say that a number $n$ collapses when powers of $n$ have unexpectedly low integer complexity. The conjecture is stated in the context of treating $n^k$ as random.

Collapse conjecture. If

$$\bar{f}_{\log}(n)<\alpha_{ave},$$

then $n$ is unlikely to collapse.

This claim appears inside an ignored passage and is presented heuristically rather than as a proved theorem. The source supplies no resolution, so it remains open.