Model 1 conjecture on the limiting pseudospectrum

Let $n$ be the matrix size, let $m\geq 1$ be fixed, and let $S_{\delta J}^{(1)}(m)$ denote model 1. Let $\lambda_0$ be the distinguished point included in the pseudospectrum, and let $f_{\widehat{S}^m}(\mathbb{T})=\mathbb{T}$ be the unit-circle symbol curve. Model 1 pseudospectrum conjecture. At each fixed $m\geq 1$, the boundary of the pseudospectrum containing $\lambda_0$ increases in size as $n$ increases, converges to the unit circle $\mathbb{T}=f_{\widehat{S}^m}(\mathbb{T})$ as $n\to\infty$, and its interior becomes filled by eigenvalues of the perturbed system as $n\to\infty$. This predicts a nontrivial limiting pseudospectral region for model 1, although the supplied text gives no proof or resolution.