On a question of Schmidt and Summerer concerning $3$-systems

Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper $3$-system $(P_{1},P_{2},P_{3})$ with the property $\overline{\varphi}_{3}=1$. In fact, our method generalizes to provide $n$-systems with $\overline{\varphi}_{n}=1$, for arbitrary $n\geq 3$. We visualize our constructions with graphics. We further present explicit examples of numbers $\xi_{1}, \ldots, \xi_{n-1}$ that induce the $n$-systems in question.