Classical large-sample consistency conjecture for the proposed signal-count estimator

Let ${\bf R}$ be an $n\times n$ covariance matrix satisfying the hypothesis of the paper's spiked-convergence proposition, and let $\widehat{{\bf R}}$ be a sample covariance matrix formed from $m$ snapshots. Write $\widehat{k}$ for the estimate of the number of signals obtained from the proposed estimator.

Classical consistency conjecture. With $n$ fixed and $m\to\infty$, $\widehat{k}$ is a consistent estimator of the true number of signals $k$.

This is the classical large-sample regime, in which consistency is proposed as a theoretical justification for the estimator. The paper presents the claim as conjectural and does not provide a proof in the supplied text.