Flip graph of point set

Flip graph of point set

Is the flip graph of every point set in $\mathbb R^3$ connected? If
not, is there a set with an isolated node?
Def: For a point set $S$, the flip graph of $S$ is a graph whose
nodes are the set of triangulations of $S$. Two nodes $T_1$ and $T_2$ of
the flip graph are connected by an arc if one diagonal of $T_1$ can
be flipped to obtain $T_2$.
Any idea would be appreciate.
These tags not available for me : flip-graph