Existence and construction of infinite well-rounded families

Determine whether an infinite family of well-rounded log unit lattices exists and, if so, construct such a family explicitly.

Background

The paper analyzes orthogonality and well-roundedness for several rank-2 families. In particular, it constructs infinite families with orthogonal log unit lattices and exhibits a non-CM $D_6$ family whose shapes converge to a well-rounded shape even though no member is well-rounded.

These results leave unresolved whether well-roundedness itself occurs infinitely often among log unit lattices. The authors explicitly ask both for existence and for an explicit construction if such a family exists.