The density conjecture for {-1,1}-perfect numbers

Let $\mathcal{A}$ denote the density of the abundant positive integers, and let the density of the positive $\{-1,1\}$-perfect numbers mean the corresponding natural density when it exists. Density conjecture for $\{-1,1\}$-perfect numbers. The positive $\{-1,1\}$-perfect numbers have density equal to $\mathcal{A}$, where

$$0.2474 < \mathcal{A} < 0.2480.$$

The conjecture would imply that the abundant numbers which are not $\{-1,1\}$-perfect have density zero, but the paper does not establish this equality.