Toth's zero Schnirelmann-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 $x$. For a set of nonnegative integers, its Schnirelmann density is the infimum of $\pi_{\mathcal{S}}(n)/n$ over positive integers $n$, where $\pi_{\mathcal{S}}(n)$ counts elements of $\mathcal{S}$ not exceeding $n$. Toth's zero Schnirelmann-density conjecture. The Schnirelmann density of $\mathcal{S}$ is $0$. The conjecture is motivated by the observed tendency of the ordinary density ratio toward $0$, despite $1\in\mathcal{S}$; no proof is given.