Prove that every harmonic divisor number is semiperfect (eq there are no weird harmonic divisor numbers)

Prove that every harmonic divisor number is semiperfect (also called pseudoperfect). A harmonic divisor number is an integer $n$ such that $n\dfrac{\sigma_0(n)}{\sigma_1(n)}$ is an integer, and a semiperfect number is an integer which can be expressed as a sum of distinct proper divisors of itself. If a semiperfect number has no semiperfect proper divisors then it's called primitive semiperfect.

Every perfect number is both a harmonic divisor number, which is easy to prove, and a primitive semiperfect number, which follows from the definition and the fact that any number with a semiperfect proper divisor must be abundant ($\sigma(n)>2n$). However, it doesn't appear to be obvious whether every harmonic divisor number is semiperfect.

I became interested in this question following this line of reasoning: It is known that no power of a prime can be a harmonic divisor number. It's also known that every practical number (one for which the distinct proper divisors can sum to every smaller integer) that isn't a power of two (which are all practical) is an even semiperfect number. Therefore, every harmonic divisor number is a practical number $\implies$ every harmonic divisor number is an even semiperfect number. Proving the statement that every harmonic divisor number is a semiperfect number, however, does not require proving the non-existence of odd perfect or odd harmonic divisor numbers, as there are infinitely many odd primitive semiperfect numbers.