A hole is chordless cycle that length of the cycle is four or more.

In this post I asked: What is the maximum number of holes that a simple graph on n vertices can have?

Gil Kalai answered that there is no polynomial upper bound.

I need a polynomial upper bound for number of holes over following class of graphs: Graphs constructed from triangles such that no two triangles have more than one vertex in common. this graphs are not necessarily chordal, and may have holes.

May I hope for a polynomial upper bound for number of holes of such graphs?