Barja–Stoppino slope conjecture for fibered varieties

Let $f:X\to B$ be a fibered $n$-dimensional variety whose relative canonical sheaf $\omega_{X/B}$ is relatively nef and ample on general fibers, and whose general fibers have sufficiently mild singularities. Writing $F$ for a general fiber and $p_g(F)$ for its geometric genus, the slope inequality is

$$K_{X/B}^n\ge n\frac{K_F^{n-1}}{p_g(F)}\deg f_*\omega_{X/B}.$$

Barja–Stoppino slope conjecture. Under these hypotheses, $f$ satisfies the slope inequality.

The conjecture extends the slope inequality proved by Cornalba–Harris and Xiao for fibered surfaces to higher-dimensional fibered varieties. The phrase “sufficiently mild singularities” is part of the stated hypothesis; the general higher-dimensional case is not resolved by the supplied source.