Exponential decay conjecture for incomplete selected inversion errors

Let $H$ be the matrix and let $E$ denote the error matrix in the notation of the paper's selected-inversion setup. Write $\mathrm{level}(i,j)$ for the level of fill 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,

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

This conjecture would quantify exponential decay of the entries discarded by incomplete selected inversion. The source presents it as an expected result and does not provide a proof; the required assumptions on the matrix and the precise meaning of the bound should be checked in the surrounding notation.