Lutz–Nevo collapse conjecture for flag triangulations of the 3-sphere

Let $T$ be a flag triangulation of $S^3$ with $\gamma_2(T)=0$. A collapse as in the source is a collapse of a 1-simplex $e$ that is not contained in any square and whose link is a square, producing another flag triangulation of $S^3$.

The three-dimensional case of Lutz and Nevo's conjecture. For every flag triangulation $T$ of $S^3$ with $\gamma_2(T)=0$, there is a sequence of such collapses that reduces $T$ to the octahedral 3-sphere.

This is presented as a conjecture of Lutz and Nevo and would reduce all flag triangulations of $S^3$ with vanishing $\gamma_2$ to the octahedral triangulation. The source provides no resolution of the conjecture.