Asymptotic maximal-eigenvalue conjecture for Haar-wavelet matrices

Let $A(N,K)$ be the $(N+1)K\times(N+1)K$ matrix whose entries are the integrals of products of Haar wavelets defined in the paper, and let $\lambda_{\max}(A(N,K))$ denote its maximal eigenvalue. Maximal-eigenvalue conjecture. Given a small $\delta>0$, sufficiently large $N,K\geq1$ can be found such that

$$\pi-\delta<\lambda_{\max}\big(A(N,K)\big).$$

In particular, for $\eta\in(\sqrt 2-1,1)$ sufficiently close to $1$, sufficiently large $N,K\geq1$ can be found such that

$$\frac{2\pi\eta}{1+\eta^2}\leq\lambda_{\max}\big(A(N,K)\big).$$

This conjecture is intended to establish that the Haar-wavelet construction yields Bell-CHSH violations arbitrarily close to Tsirelson's bound for the free massless $(1+1)$-dimensional spinor field. The paper provides numerical and asymptotic evidence, but the asserted eigenvalue bound is not proved.