Pliability conjecture for terminal factorial quartic hypersurfaces

Let $X\subset \mathbb{P}^4$ be a terminal factorial quartic hypersurface. Let $\mathcal P(X)$ denote its pliability, namely the set of square-birational-equivalence classes of Mori fibre spaces birational to $X$; write $[X]$ for the class represented by $X$.

Pliability conjecture. The set $\mathcal P(X)$ is finite, and

$$\mathcal P(X)=\{[X]\}$$

precisely when $X$ has no worse than $cA_1$ singularities. In particular, no terminal factorial quartic hypersurface is rational.

This general conjecture extends earlier work cited by the authors. The supplied text does not give evidence of a resolution.