Local limit conjecture for the simplified polymer model

Let $s_*$ be the location of the maximum of $Y$, let $c_h$ be the model-dependent centering constant, and let $\widetilde{\mathbf{B}}$ be the two-sided three-dimensional Bessel process from the Bessel coupling conjecture. Define

$$s_*^n=s_*n^{2/9}-\lfloor s_*n^{2/9}\rfloor, \qquad \theta_\omega(n)=\sum_{k\in\mathbb{Z}}e^{-\beta\mathcal{W}_{s_*^n+k}(\omega)}.$$

Simplified-model local limit conjecture. For all sufficiently large $n$, for the upper extremal point $M_n^+$ and every integer $k$,

$$\mathbf{P}_{n,h}^{\omega,\beta}\left(M_n^+=c_hn^{1/3}+\lfloor s_*n^{2/9}\rfloor+k\right)\sim\frac{1}{\theta_\omega(n)}e^{-\beta\mathcal{W}_{s_*^n+k}}.$$

This conjecture describes the fourth-order, order-one local fluctuations in the simplified model. It is intended to follow from the Bessel coupling conjecture together with the exclusion of trajectories outside the relevant window, but remains unproved.