Sparsity conjecture for the cryo-EM covariance matrix blocks

Let $\hat L^{k_1,k_2}$ be a block of the matrix $\hat L$ arising in the covariance-based cryo-electron microscopy algorithm, with $k_1,k_2$ its associated frequency indices. Let $\operatorname{nnz}(A)$ denote the number of nonzero entries of a matrix $A$. Sparsity conjecture. The number of nonzero entries satisfies

$$\operatorname{nnz}(\hat L^{k_1,k_2})\leq \frac{1}{k_1+k_2+1}\left(\frac{(k_1+1)(k_1+2)(k_2+1)(k_2+2)}{4}\right)^2.$$

The squared factor is the total number of entries in $\hat L^{k_1,k_2}$, so the conjecture asserts that the proportion of nonzero entries decreases with the frequencies. Numerical experiments support this sparsity pattern, but the paper states that it remains to be verified theoretically; it leads to the claimed $O(N_{\mathrm{res}}^7)$ storage complexity for the largest block.