Pliability of Mori fibre spaces is an algebraic variety

Let $X\to S$ be a Mori fibre space, and let $\mathcal{P}(X/S)$ denote its pliability. Pliability algebraicity conjecture. The pliability $\mathcal{P}(X/S)$ is, in a natural way, an algebraic variety. The conjecture proposes an algebro-geometric structure on the set of Mori fibre space models birational to $X/S$.