Exponential decay conjecture for incomplete inversion errors

Let $A$ be the matrix, let $B$ be the output of incomplete selected inversion, and let $F$ denote the error matrix in the notation of the paper's selected-inversion setup. Write $d(i,j)$ for the relevant graph distance between indices $i$ and $j$, let $g_\mathcal{E}(z)$ be the decay-rate function, and write $\lesssim_\varepsilon$ for an inequality up to a constant depending on $\varepsilon$. Exponential decay conjecture. In the notation of the selected-inversion setup,

$$|F(i,j)| \lesssim_\varepsilon \exp\bigl(-g_\mathcal{E}(z)\,d(i,j)\bigr).$$

This conjecture asserts spatial exponential localization of the error produced by incomplete selected inversion. The surrounding argument verifies the form of the estimate only in a simplified case and indicates that the general estimate would require repeated use of the affine error-propagation relation; no general proof is supplied.