Exact formula conjecture for the symmetric subset function

Let $\Delta(\varepsilon)$ denote the symmetric subset function studied in the source, with $0\leq\varepsilon\leq1$. Exact formula conjecture for $\Delta$.

$$\Delta(\varepsilon)=\max\left\{2\varepsilon-1,\frac\pi4\varepsilon^2\right\}$$

for all $0\leq\varepsilon\leq1$.

The source presents exact determination of $\Delta(\varepsilon)$ as the principal remaining question. It has established bounds on $\Delta$, but the exact value throughout the interval remains open in the supplied text.