Discrete ALIF inner-loop convergence conjecture under the spectral condition

Let $K_n$ be the Discrete ALIF iteration matrices, and assume the hypotheses of the preceding spectral-distribution lemma: either $L(x)$ is a step function, or $k(x)$ and $L(x)$ are continuous with $L(x)\geq L_*>0$. Define

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

The Discrete ALIF inner-loop convergence conjecture asserts that if

$$0\leq\kappa(x,\theta)\leq 2,\qquad (x,\theta)\in[0,1]\times[-\pi,\pi],$$

then the inner loop of Discrete ALIF converges. The conjecture is a reinterpretation of the full-method claim after examples show that the spectral symbol can control the inner loop without guaranteeing convergence of the complete ALIF method; determining sufficient conditions for the full method remains separate.