The non-smooth curvature-dimension conjecture for warped products

Let $(B,d_B)$ be a complete Alexandrov space with $\dim_B=d$, and let $(F,d_F,m_F)$ be a metric measure space. Let $f:B\rightarrow\mathbb{R}_{\geq 0}$ be continuous with $\partial B\subset f^{-1}(\{0\})$. Assume that $(F,m_F)$ satisfies $CD((N-1)K_F,N)$ and that $f$ is $\mathcal{F}K$-concave, with the following conditions:

1. If $\partial B=\emptyset$, suppose $K_F\geq Kf^2$.
2. If $\partial B\neq\emptyset$, suppose $K_F\geq 0$ and $Df_p\leq\sqrt{K_F}$ for all $p\in X$.

Non-smooth warped-product conjecture. Then the $N$-warped product $B\times^N_f F$ satisfies $CD((N+d-1)K,N+d)$.

The theorem preceding this statement establishes the corresponding result in the smooth setting, while the singularity-transport result applies when $B$ is Alexandrov and $F$ is a general metric measure space. The conjecture proposes that the curvature-dimension bound extends to this non-smooth warped-product setting.