The polygonal formulas for Delta three and Delta four

For $k\geq1$, let $\Delta_k(\varepsilon)$ denote the function studied in the paper, with $0\leq\varepsilon\leq1$. Delta three and Delta four polygonal conjecture. The graph of $\Delta_3(\varepsilon)$ is the polygonal path connecting

$$(0,0),\left(\frac47,\frac27\right),\left(\frac7{11},\frac4{11}\right),\left(\frac57,\frac37\right),(1,1),$$

and the graph of $\Delta_4(\varepsilon)$ is the polygonal path connecting

$$(0,0),\left(\frac5{12},\frac16\right),\left(\frac9{19},\frac4{19}\right),\left(\frac12,\frac29\right),\left(\frac23,\frac{10}{27}\right),\left(\frac{17}{24},\frac5{12}\right),(1,1).$$

The authors report computational evidence that these are upper bounds for the corresponding graphs, but have not verified that the bounds are sharp.