Entropy criterion for finiteness of the weak-disorder critical point

Let $K(n)$ be the distribution of the renewal increments, and let $\beta_c$ be the critical value separating weak disorder from strong disorder. Assume that the environment is unbounded from above.

Entropy criterion conjecture. The weak-disorder critical point is finite if and only if

$$\beta_c<\infty\quad\Longleftrightarrow\quad\sum_{n\geq1}K(n)\log\frac{1}{K(n)}<\infty.$$

The criterion is proposed as a characterization of when a strong-disorder phase can occur in the heavy-tailed directed-polymer model. The surrounding discussion presents it as an unresolved question, particularly in the regime where the tail of $K$ is near the entropy threshold; no resolution is supplied in the source.