The higher-dimensional slope inequality for fibred varieties

Let $g\colon Y\longrightarrow T$ be a fibred variety such that the relative canonical sheaf $\omega_g$ is relatively nef and ample on the general fibres, and such that the general fibres have sufficiently mild singularities. The slope inequality is

$$K_g^n\geq n\frac{K_F^{n-1}}{p_g(F)}\deg g_*\omega_g.$$

Higher-dimensional slope inequality. Under these hypotheses, the fibred variety satisfies the slope inequality. This is known for relatively minimal fibred surfaces of genus at least $2$, while the corresponding assertion in higher dimensions is presented as a general expectation.