Bessel coupling conjecture for the local behavior at the Brownian maximum

Condition on $s_*>0$, the location of the maximum of the process $Y$, and set $\alpha=2c_{h,\beta}s_*$. For $s>0$, define the one-sided comparison processes $R_s^\pm=B_s\pm\alpha s$ and write $R=R^-\mathbbm{1}_{\mathbb{R}^-}+R^+\mathbbm{1}_{\mathbb{R}^+}$. Let $Y$ be the relevant Brownian-motion-with-parabolic-drift process, and let $\widetilde{\mathbf{B}}$ be a two-sided three-dimensional Bessel process.

Bessel coupling conjecture. There exists a coupling of $(R,Y)$ and $\widetilde{\mathbf{B}}$ such that almost surely there are $\delta_1>0$ and $n_1\in\mathbb{N}$ for which, for every $n\geq n_1$ and every $|u|<\delta_1$,

$$n^{1/9}R_{un^{-2/9}}=n^{1/9}\left(Y_{s_*}-Y_{s_*+u/n^{2/9}}\right)=\widetilde{\mathbf{B}}_u.$$

The conjecture is motivated by the fact that Brownian motion with drift conditioned to remain positive converges to a three-dimensional Bessel process. Establishing this exact local coupling would provide the almost-sure control needed for the finer local-limit analysis of the simplified polymer model.