Kravitz–Sah conjecture on the minimum size of posets with n linear extensions

Let $\mu(n)$ denote the minimum number of elements in a finite poset with exactly $n$ linear extensions. Kravitz–Sah conjecture.

$$\mu(n)=O(\log n).$$

This would improve the known bound $\mu(n)=O(\log n\,\log\log n)$ and would give substantially smaller posets realizing prescribed numbers of linear extensions.