Kaczynski's vanishing-area conjecture for unrestricted trapezoids

Let $P_{j,k}^n$ be the parallelograms defining an unrestricted trapezoid $T^n_\sigma$ for a permutation $\sigma\in\operatorname{Sym}(n)$, and let

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

Kaczynski's vanishing-area conjecture.

$$\lim_{n\rightarrow\infty}\alpha(n)=0.$$

Kaczynski claimed to have proved this, and the paper states that it follows from Proposition 1; thus the conjecture is resolved in the source.