Condition-number conjecture for cryo-EM covariance matrix blocks

For frequency indices $k_1$ and $k_2$, let $\hat L^{k_1,k_2}$ be the corresponding block of $\hat L$, and write $\kappa(\hat L^{k_1,k_2})$ for its condition number. Condition-number conjecture.

$$\kappa(\hat L^{k_1,k_2})\leq 1.4818+0.8524\min(k_1,k_2).$$

This follows heuristically from the proven lower bound $\lambda_{\min}(\hat L^{k_1,k_2})\geq 1/(2\pi)$ together with the preceding conjectured linear growth of the maximal eigenvalue. The bound remains conjectural because the maximal-eigenvalue estimate is supported by numerical experiments rather than a theoretical proof.