Generalized-cone characterization of the timelike curvature-dimension condition

Let $N\in(1,\infty)$, $\kappa\in\mathbb{R}$, and let $X$ be a proper, complete, geodesic metric space with a Radon measure $\mathfrak{m}$. Let $f:I\to[0,\infty)$ be Lipschitz and satisfy $f^{-1}(\{0\})=\partial I$. Write $Y={}^-\!I\times_f^N X$ and set

$$\eta:=\sup_I\{-(f')^2+\kappa f^2\}.$$

Generalized-cone characterization conjecture. The space $Y$ satisfies $\textnormal{\textsf{TCD}}(-\kappa N,N+1)$ if and only if $f”-\kappa f\leq0$ in the distributional sense in $I$, and $X$ satisfies $\textnormal{\textsf{CD}}(\eta(N-1),N)$. This is proposed as a generalization of the proved warped-product results in the paper; the full if-and-only-if characterization remains open.