Critical-moment growth conjecture for directed polymers

Let $W_n^\beta$ be the normalized partition function at time $n$, let $p^*(\beta)$ be the critical moment exponent, and suppose weak disorder holds. Define

$$\kappa(\beta):=\min\left(1,\frac{d}{2}\bigl(p^*(\beta)-1\bigr)-1\right)\in[0,1].$$

Critical-moment growth conjecture. The critical moment satisfies

$${{\mathbb E}}\left[(W_n^\beta)^{p^*(\beta)}\right]=n^{\kappa(\beta)+o(1)}\quad\text{as }n\to\infty.$$

The conjecture seeks a sharper description than the currently established at-most-linear growth and connects the critical moment to the critical behavior of the model. The source gives no resolution.