The exact value of the normalized nine-point quadrilateral area

Let $\Delta_4'(n)$ denote the minimum, over configurations of $n$ points in the normalized setting considered in the paper, of the area of a convex quadrilateral containing four points. The figure gives the lower bound $\Delta_4'(9)\geq 1/4$. The nine-point quadrilateral conjecture.

$$\Delta_4'(9)=1/4.$$

Equivalently, the other direction $\Delta_4'(9)\leq 1/4$ should hold. Establishing this would determine the exact value of the nine-point case, beyond the displayed lower bound.