The nonsquare odd abundant numbers conjecture for {-1,1}-perfect numbers

Let $n$ be a positive odd integer. Call $n$ abundant if $\sigma(n) \geq 2n$, and call it nonsquare if it is not a perfect square. Nonsquare odd abundant numbers conjecture. Every nonsquare odd abundant number is $\{-1,1\}$-perfect. The paper notes that this holds for every such number checked computationally, while Theorem 3 rules out odd square abundant numbers; the conjecture remains open.