The exact value of the normalized $k$-gon density at multiples of $k-1$

Let $k$ be a constant, let $\alpha$ satisfy $\alpha\leq k-1$, and let $\Delta_k'(n)$ denote the normalized extremal area parameter for convex $k$-gons defined in the paper. The normalized $k$-gon conjecture.

$$\Delta_{k}'(\alpha(k-1))=\dfrac{1}{2(\alpha-1)}$$

for $\alpha\leq k-1$. This would give exact values at the indicated arguments, whereas the paper's general results provide upper bounds of order $(k-2)/n$ and do not establish this formula.