It is known that some triangulation graphs admit straight-line drawings realizing certain characteristics, e.g., greedy triangulation, minimum- weight triangulation, Delaunay triangulation, etc.. Lenhart and Liotta [12] in their pioneering paper on “drawable” minimum-weight triangulations raised an open problem: ‘Does every triangulation graph whose skeleton is a forest admit a minimum-weight drawing?’ In this paper, we answer this problem by disproving it in the general case and even when the skeleton is restricted to a tree or, in particular, a star.