Conjecture on error density estimation for long-memory regression errors

Consider the regression model $Y_i=\beta_0+\beta_1X_1+\varepsilon_i$ and the Parzen–Rosenblatt estimator

$$\hat f_{h,\Delta}(x)=\frac{1}{nh}\sum_{i=1}^n K_h(x-\hat\varepsilon_i).$$

Assume conditions (P) and (E), and assume that the estimator expansion in holds. Suppose that $F_{\eta}$ is five times differentiable with bounded, continuous, and integrable derivatives, that $\alpha_{\varepsilon}<1/2$, and that

$$nh^5\to 0,\qquad \sigma_{n,2}h\to\infty.$$

Error-density estimation conjecture. For each fixed $x$,

$$\frac{n}{\sigma_{n,2}}\left(\hat f_{h,\Delta}(x)-f(x)\right)\xrightarrow{d} f_{\varepsilon}^{(2)}(x)\left(Z_2-\frac{1}{2}Z_1^2\right).$$

This asserts a nonstandard limiting distribution for residual-based error-density estimation in the long-memory regime $\alpha_{\varepsilon}<1/2$. The supplied text gives the claimed limit but no evidence of whether it has been proved or remains open.