Suppose you are given a 3-ball $B$ in $\mathbb{R}^3$ that is bounded by a PL sphere, a triangulation $T$ of $B$ by Euclidean tetrahedra. Is that triangulation necessarily shellable?

I know that if $T$ can be lifted to a convex hypersurface in $\mathbb{R}^4$, then it is shellable; this applies if $T$ comes from a Delaunay triangulation. But I believe most triangulations are not liftable.

I suspect the answer is "no", that $T$ is not necessarily shellable. The first obstruction I found was the existence of a knotted arc made out of two edges of the triangulation; this obviously cannot happen for triangulations of the kind that I described.