Generalized GLT spectral-distribution conjecture for ALIF iteration matrices

Let $k$ be an even, non-negative, bounded, compactly supported measurable function with $\\|k\\|_1=1$, supported on $[-1,1]$, and let $L(x)$ be strictly positive on $[0,1]$. Define

$$f_j(x):=\frac{k(j/L(x))}{L(x)}.$$

Suppose that $f_j$ is Riemann-integrable over $[0,1]$ for every $j\in\mathbb Z$. For

$$\kappa(x,\theta):=\frac{1}{L(x)}\sum_{j\in\mathbb Z}k\left(\frac{j}{L(x)}\right)e^{\operatorname{i}j\theta},$$

and the ALIF iteration matrices $K_n$, the generalized GLT spectral-distribution conjecture asserts that

$$\{K_n\}_n\sim_{\mathrm{GLT},\sigma,\lambda}\kappa(x,\theta).$$

This conjecture would extend the known result for step-function $L$ or continuous $k$ and $L$ with $L(x)\geq L_*>0$; it concerns the asymptotic spectral and singular-value distribution of the ALIF iteration matrices under the weaker Riemann-integrability hypothesis.