Very strong disorder conjecture for quenched and annealed free energies

Assume (Dir), (Ind), (Loc), $V\in\mathcal{L}$, and $\Lambda_q(V)<\infty$. Let $\underline h_\infty^\lambda$ and $\bar h_\infty^\lambda$ denote the lower and upper limiting free-energy quantities, respectively, and let very strong disorder be understood in the model's standard sense. Very strong disorder conjecture. For $\lambda=\Lambda_q(V)$, whenever there is very strong disorder,

$$\mathbb{P}\bigl(0 = \underline h_\infty^\lambda< \bar h_\infty^\lambda= \infty \bigr) = 1.$$

This conjecture describes the extreme separation between the lower and upper limiting free energies in the very strong disorder regime. The surrounding discussion notes that very strong disorder occurs in dimension two and that minimizers of the relevant variational formulas need not exist in this regime; the asserted almost-sure behavior remains the conjectural part.