Erdős Problem #18 — Representing integers below $n!$ as sums of few distinct divisors

I proved long ago that every $m < n!$ is the distinct sum of $n - 1$ or fewer divisors of $n!$. Let $h(m)$ be the smallest integer, if it exists, for which every integer less than $m$ is the distinct sum of $h(m)$ or fewer divisors of $m$. Srinivasan called the numbers for which $h(m)$ exists practical. It is well known and easy to see that almost all numbers $m$ are not practical. I conjectured that there is a constant $c \geq 1$ for which for infinitely many $m$ we have $h(m) < (\log\log m)^c$. M. Vose proved that $h(n!) < cn^{1/2}$. Perhaps $h(n!) < c(\log n)^{c_2}$. I would be very glad to see a proof of $h(n!) < n^{\varepsilon}$.