More specifically, given two handle decompositions of a PL (relative) cobordism $W$, is it always possible to go from one handle decomposition to the other via a sequence of handle slides and handle cancellations?

I think I have an argument, but I wanted to know if it is already known (and also check my argument): Choose some smoothing of the cobordism $W$. Construct Morse functions $f_0$ and $f_1$ that give the two handle decompositions (but smoothed). Find a homotopy $f_t$ that interpolates them such that $f_t$ only has at worst birth-death singularities - then $f_t$ gives a corresponding set of moves between handle decompositions. Approximate $W \times I$ by a triangulation, and approximate $f_t$ by a PL map. This should give the sequence of handle moves in PL.

Is there an analog of Cerf theory in PL?

More specifically, given two handle decompositions of a PL (relative) cobordism $W$, is it always possible to go from one handle decomposition to the other via a sequence of handle slides and handle cancellations?

I think I have an argument, but I wanted to know if it is already known (and also check my argument): Choose some smoothing of the cobordism $W$. Construct Morse functions $f_0$ and $f_1$ that give the two handle decompositions (but smoothed). Find a homotopy $f_t$ that interpolates them such that $f_t$ only has at worst birth-death singularities - then $f_t$ gives a corresponding set of moves between handle decompositions. Approximate $W \times I$ by a triangulation, and approximate $f_t$ by a PL map. This should give the sequence of handle moves in PL.