Kaczynski's monotonicity conjecture for unrestricted trapezoid areas

For each positive integer $n$, let

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

where $T^n_\sigma$ is the smallest-area unrestricted trapezoid associated with a permutation of $n$ parallelograms. Kaczynski's monotonicity conjecture. The sequence $\alpha(n)$ decreases monotonically. The source presents this as one of Kaczynski's open questions; no proof or disproof is supplied there.