Conjectured distributional limit for the quenched CLT error in one-dimensional random environments

Let $F_{n,\omega}$ be the quenched distribution function associated with the hitting time $T_n$ of a one-dimensional random walk in a random environment, and let $\Phi$ denote the standard normal distribution function. For $\kappa\in(2,3)$, consider the rescaled quenched central limit theorem error

$$n^{\frac{3}{2}-\frac{3}{\kappa}}\left\|F_{n,\omega}-\Phi\right\|_\infty.$$

Distributional-limit conjecture. If $\kappa\in(2,3)$, then

$$n^{\frac{3}{2}-\frac{3}{\kappa}}\left\|F_{n,\omega}-\Phi\right\|_\infty$$

converges in distribution to a non-degenerate random variable on $(0,\infty)$. Theorem gives the corresponding order of the maximal quenched CLT error, but the multiplicative factor oscillates between $0$ and $\infty$ depending on $\omega$ and $n$; identifying a limiting distribution remains open.