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 ${\bar{\phi }}_3=1$. In fact, our method generalizes to provide $n$-systems with ${\bar{\phi }}_n=1$, for arbitrary $n\ge 3$. We visualize our constructions with graphics. We further present explicit examples of numbers ${\xi }_1,\dots ,{\xi }_{n-1}$ that induce the $n$-systems in question.