Linear-growth conjecture for maximal eigenvalues of cryo-EM covariance blocks

For frequency indices $k_1$ and $k_2$, let $\hat L^{k_1,k_2}$ denote the corresponding block of the matrix $\hat L$, and let $\lambda_{\max}(\hat L^{k_1,k_2})$ be its maximal eigenvalue. Maximal-eigenvalue growth conjecture. The maximal eigenvalue of $\hat L^{k_1,k_2}$ grows linearly with $\min(k_1,k_2)$. Numerical experiments show a clear linear dependence in the case $k_1=k_2=k$, with an approximate fit $0.2358+0.1357k$. Together with the proven lower bound on the minimal eigenvalue, this conjecture would yield a linear condition-number bound, but the required upper bound on the maximal eigenvalue is not proved.