Kaczynski's logarithmic-area conjecture for unrestricted trapezoids

For the smallest unrestricted trapezoid area

$$\alpha(n)=\min_{\sigma\in\operatorname{Sym}(n)}\mu_2(T^n_\sigma),$$

where $T^n_\sigma$ is the union of the parallelograms associated with $\sigma$, Kaczynski's logarithmic-area conjecture. there exist constants $c_\pm>0$ such that

$$\frac{c_-}{\log n}<\alpha(n)<\frac{c_+}{\log n}.$$

Kaczynski claimed that the lower-bound constant exists and gave a weaker upper bound; the source leaves this sharper two-sided estimate open.