K-stable-fibre conjecture for smooth uniruled varieties

Let $X$ be a smooth uniruled variety. K-stable-fibre conjecture. The variety $X$ is birational to a Mori fibre space whose general fibre is K-stable. This conjecture links birational classification of uniruled varieties with K-stability: it predicts a birational Mori fibre structure whose general fibres have the strongest stability property relevant to the survey. The source attributes the conjecture to the cited literature but gives no resolution, so it remains open here.