Semi-Riemannian Bakry–Émery warped-product formula

Let $B\times_f^{N}F$ be an $N$-warped product of a Riemannian manifold $B$, a positive warping function $f$, and a weighted Riemannian manifold $F$. The weighted semi-Riemannian analogue is obtained by replacing the Riemannian warped-product geometry with a weighted semi-Riemannian one. Semi-Riemannian Bakry–Émery formula conjecture. A formula like the Bakry–Émery $N$-Ricci warped-product formula holds in the context of weighted semi-Riemannian manifolds. This would extend the displayed Riemannian formula to weighted Lorentzian and more general semi-Riemannian settings; no proof or precise general formulation is supplied here.