Connectedness conjecture for the lattice of superabundant numbers

Let $\mathcal{S}$ be the set of superabundant numbers, meaning the integers $n$ for which

$$\frac{\sigma(n)}{n}>\frac{\sigma(m)}{\sigma(m)}\qquad\text{for }1\leq m<n,$$

where $\sigma(n)$ denotes the sum of the divisors of $n$. Consider the lattice whose vertices are the elements of $\mathcal{S}$ and whose links correspond to multiplication or division by primes.

Connectedness conjecture. Given a superabundant number $n$, either there is a prime $p$ such that $np$ is superabundant or there is a prime $q$ such that $n/q$ is superabundant; moreover, there is always a chain of primes $p_1,p_2,\ldots,p_m$ such that successive multiplication or division by these primes reaches $1$.

This is proposed as a weaker version of the Alaoglu–Erdős conjecture after computational counterexamples to both its upward and downward halves. The source does not report a resolution of this weaker conjecture.