Existence of W_{3,4}-free triangulations on ten or eleven vertices

Determine whether there exists a W_{3,4}-free plane triangulation on ten or eleven vertices.

Background

The paper proves that every W_{3,4}-free plane triangulation has at most eleven vertices. Since W_{3,4} itself has ten vertices, triangulations with at most nine vertices are automatically W_{3,4}-free. The authors do not resolve whether the upper range of ten or eleven vertices is attainable.