The strict inequality conjecture for the polymer moment threshold

Let $p^*(\beta)$ denote the critical moment threshold for the directed polymer model, and let $d$ be the spatial dimension. The weak-disorder critical inverse temperature is denoted by $\beta_{cr}$.

Strict inequality conjecture. There exists no $\beta\in\mathbb R_+$ such that

$$p^*(\beta)=1+2/d.$$

This is a reformulation of the widely held conjecture that the strong-disorder criterion holds at $\beta_{cr}$. If true, it implies that $p^*(\beta)>1+2/d$ throughout the weak-disorder phase.