Topological invariance of pliability for threefold Mori fibre spaces

Let $X/S$ be a threefold Mori fibre space, and let $\mathcal{P}(X/S)$ denote its pliability. Topological pliability conjecture. The pliability $\mathcal{P}(X/S)$ is, in a natural way, a topological invariant. This would make the birational models represented by pliability depend only on the topology of the total space in this setting.