Toth's zero asymptotic-density conjecture for spoof odd perfect numbers

Let $\mathcal{S}$ be the set of odd positive integers $n$ such that $2n/\sigma(n)-1=1/x$ for some positive integer $x$. Its asymptotic density is the limit, when it exists, of

$$\frac{\pi_{\mathcal{S}}(n)}{n},$$

where $\pi_{\mathcal{S}}(n)$ counts elements of $\mathcal{S}$ not exceeding $n$. Toth's zero asymptotic-density conjecture. The asymptotic density of $\mathcal{S}$ is $0$. This conjecture expresses the observed scarcity of such integers; the paper presents numerical evidence but does not establish the assertion.