Graded curvature-dimension version of the contraction and spectral conjectures

Let a weighted manifold be diffeomorphic to $\mathbb{R}^n$ and satisfy the separate curvature bounds

$$\operatorname{Ric}_g\geq0,\qquad \nabla_g^2V\geq\rho g.$$

Here $\operatorname{Ric}_g$ is the Riemannian Ricci tensor and $\nabla_g^2V$ is the Hessian of the weight potential $V$. Graded curvature-dimension conjecture. The tentative contraction and spectral comparison conjectures hold true when restricted to weighted manifolds satisfying these bounds and still diffeomorphic to $\mathbb{R}^n$. The proposed restriction is intended to replace a single lower bound on the weighted Ricci tensor by separate bounds on its geometric and potential components. The graded curvature-dimension condition is to be introduced in future work, and the source gives no resolution.