A harmonic divisor number or Ore number is a positive integer whose harmonic mean of its divisors is an integer. In other words, $n$ is a harmonic divisor number if and only if $\dfrac{nd(n)}{\sigma(n)}$ is an integer, where $d(n)$ is the number of divisors of $n$ and $\sigma(n)$ the sum of divisors of $n$. For example, for $n=140$, we have $\dfrac{nd(n)}{\sigma(n)}=\dfrac{140\times 12}{336}=5$, so $140$ is a harmonic divisor number. I was wondering:
Is there any square harmonic divisor number greater than $1$?
I check the conjectured list of the first $10000$ harmonic divisor numbers and found no squares. In the OEIS page it is conjectured that every harmonic divisor number greater than $1$ is a Zumkeller number, one whose divisors can be partitioned into two disjoint sets with equal sum. Of course no square number can be a Zumkeller number because $\sigma(n)$ is odd for square $n$, but perhaps proving this conjecture is even harder. So I was wondering if we could prove there is no square harmonic divisor number greater than $1$ or, in case where this seems to be equally difficult as finding an odd perfect number, provide any lower bound for a square harmonic divisor number? Any help/reference appreciated.