Lorentzian warped-product timelike curvature equivalence

Let $I$ be an interval, let $F$ be a metric space, and let $f$ be the warping function in the Lorentzian warped product $-I\times_fF$. Write

$$K_F=\sup\{-(f')^2+Kf^2\}.$$

Lorentzian warped-product curvature conjecture. The Lorentzian warped product $-I\times_fF$ has timelike CBB by $K$ if and only if $f$ is $-Kf$-concave and $F$ has CBB by $K_F$. The forward implication is given as a theorem immediately before the conjecture; the conjectural content is the converse, whose status is not resolved in the source.