Conjecture 1. Are there infinitely many practical numbers $m$ such that $h(m) < (\log \log m)^{O(1)}$?

$h(6) = 2$: divisors are {1, 2, 3, 6}. The hardest m to represent is m=4 or m=5, each requiring 2 divisors: 4=1+3, 5=2+3.