Determine the true growth of the divisor-sum complexity of factorials

Determine whether the factorial divisor-sum complexity satisfies h(n!)<n^{o(1)}, or even h(n!)<(log n)^{O(1)}, where h(n!) is the least number of distinct divisors of n! needed to represent every integer from 1 to n! as a sum.

Background

The paper defines h(N) as the smallest integer k such that every integer between 1 and N can be represented as a sum of at most k distinct divisors of N. Its main theorem proves the upper bound h(n!)\le(2\log 2+o(1))n/\log n, while a counting argument yields the lower bound h(n!)\gg(\log n)^2. Thus, a substantial gap remains between the known lower and upper bounds. The cited question attributed to Erdős asks whether h(n!) has subpolynomial growth, or even polylogarithmic growth, and is not resolved by the paper.