The linear-eigenvalue-growth conjecture for the Hermitian-Toeplitz matrix T

Let $N$ be the system dimension, let $\hat{\beta}$ and $\hat{\eta}$ be the receiver parameters, and let $\boldsymbol{\Psi}(\hat{\boldsymbol{\tau}}_{\hat{\mathcal{A}}})$ be the matrix formed from the selected delay estimates. Define the Hermitian-Toeplitz matrix

$$\mathbf{T}=\hat{\beta}^{-1}\hat{\eta}\,\boldsymbol{\Psi}(\hat{\boldsymbol{\tau}}_{\hat{\mathcal{A}}})\boldsymbol{\Psi}^{\mathrm{H}}(\hat{\boldsymbol{\tau}}_{\hat{\mathcal{A}}}).$$

Linear-eigenvalue-growth conjecture. There exists an upper bound on the largest eigenvalue of $\mathbf{T}$ that grows linearly with $N$:

$$\lambda_{\max}(\mathbf{T})=\mathcal{O}(N).$$

The conjecture is introduced to justify an iterative conjugate-gradient method whose iteration count, and hence computational complexity, depends on the eigenvalue behavior of $\mathbf{T}$. The source gives no resolution, so the conjecture remains open.