Localization transition conjecture for polymers in a Poissonian medium

Let $p(\beta,\nu)$ be the quenched free energy, let $\lambda(\beta)$ be the corresponding annealed free-energy function, let ${\cal D}(\nu)$ be the domain of admissible inverse temperatures, and let ${\cal L}$ denote the low-temperature/high-density region where the quenched free energy is strictly smaller than the annealed one. Localization transition conjecture. For all $\nu$,

$$\Big\{ (\nu,\beta): \beta \in {\cal D}(\nu), \frac{\partial}{\partial \beta} p(\beta, \nu) < \nu \lambda'(\beta) \Big\}={\cal L}.$$

This conjecture asserts that the region identified by the strict derivative inequality is exactly the full region of path localization, namely the region where the quenched free energy is strictly below the annealed free energy. The surrounding results establish localization under the displayed condition, but the equality with the entire region ${\cal L}$ remains conjectural in the source.