Consider an $n$-dimensional PL manifold $P$ together with a closed subpolyhedron $P_0$ such that there exists a finite combinatorial triangulation of $P$ that restricts to a triangulation of $P_0$.
Is it true that any two combinatorially equivalent combinatorial triangulations of $P$ that coincide on $P_0$ are related by a finite sequence of bistellar flips (Pachner moves) which are the identity on $P_0$?
The case of $P$ a $3$-manifold with boundary $\partial P= P_0$ is Theorem 2.1.2 of Quantum Invariants of Knots and Three Manifolds. My primary motivation for this question is to try to massage such a theorem, if it exists, to try to answer a previous MO question which I asked. But I also care about the answer for its own sake. Note that $P_0$ is not a submanifold, but only a subpolyhedron. Even if the statement isn't true in full generality, is it at least true if $P_0$ is somehow not "too pathological"?