Conjecture on fifth-order weighted central limit approximation

Let $X_1,\ldots,X_N$ be independent and identically distributed centered random variables satisfying

$$\mathbf{E}X_1=0,\qquad \mathbf{E}X_1^2=1,\qquad \mathbf{E}X_1^3=0,\qquad \mathbf{E}|X_1|^5<+\infty.$$

Let $\bar w(x)\in C_0^\infty([0,1])$ be a weighting function with

$$\int_0^1\bar w(x)\,dx=1,$$

and define

$$\bar A_N=\sum_{n=0}^{N-1}\bar w(n/N)>0.$$

Weighted fifth-order central limit conjecture. There exists such a weighting function $\bar w$ and a universal constant $C_3>0$ for which

$$\sup_{x\in\mathbb{R}}\left|\mathbf{P}\left(\frac{1}{\sqrt{\bar A_N}}\sum_{n=1}^{N-1}\sqrt{\bar w(n/N)}X_n\leq x\right)-\mathcal{G}(x)\right|\leq\frac{C_3\mathbf{E}|X_1|^5}{N^{3/2}},$$

where

$$\mathcal{G}(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x\exp(-t^2/2)\,dt-\frac{\mathbf{E}|X_1|^4-3}{8\sqrt{2\pi}N}(x^3-3x)\exp(-x^2/2).$$

This is proposed as a measure-theoretic analogue of known weighted central limit estimates with corrected normal distributions. The conjectural part is the existence of a smooth normalized weighting function satisfying the stated uniform $N^{-3/2}$ bound; the source gives no resolution.