Motivation: Let $\pi:X\to B$ be a holomorphic fibre space. By theorem 1.3 of Kawamata, if the central fibre be of the general type then all the fibres are of the general type see http://arxiv.org/pdf/math/9809091.pdf , this tells you that if the central fibre $X_0$ has Kahler-Einstein metric with negative Ricci curvature, then all the fibres have Kahler-Einstein metric with negative Ricci curvature.
Or, if the central fibre $X_0$ be a Calabi-Yau variety with canonical singularities then all the fibres $X_t$ are also Calabi-Yau variety with at worst canonical singularities. See Fibration when central fibre is a Calabi-Yau variety with canonical singularities
This means that if the central fibre has Ricci flat metric then all the fibres must have Ricci flat metric,
So it is natural to see what will happen for Fano fibration, like Mori fibre space, when the central fibre admit Kahler-Einstein metric with positive Ricci curvature and to ask all fibres $X_t$ admit Kahler-Einstein metric with positive Ricci curvature?
Question: Let $\pi:X\to B$ be a Fano fibration with Fano projective varieties $X,B$, such that $X$ is K-stable and central fibre $X_0$ admit Kahler-Einstein metric, then my conjecture is that, all the fibres must admit Kahler-Einstein metric. Is there any counter-example?
Huristicly it must be correct.
More comments with motivation of this question: Huristicly Let $f:X\to Y$ be a Fano fibration such that $X$ and central fibre $X_0$ admit Kahler-Einstein metric, then Semi-positivity of fiberwise Kahler-Einstein metric along Mori-fibre space $\pi:X\to Y$ is equivalent to K-stability of projective variety $X$ when Fano central fibre admit Kahler-Einstein metric If this is OK, then K-stability is equivalent to invariance of plurigenera due to existence of solution of "relative Kahler Ricci flow"
$$\frac{\partial \omega(t)}{\partial t}=-Ric_{X/B}(\omega(t))+\omega(t)$$
along Mori-fibre space.
