A positive eigenvalue lower bound under a Ricci lower bound

Let $(M^n,g)$ be a compact Riemannian manifold satisfying $\operatorname{Ric}(M)>K$, and let $v$ be a one-form with $\|v\|_\infty<C$. Suppose that there exists $u\in W^{2,p}(M)$ satisfying

$$\Delta u+v(\nabla u)=\lambda u$$

with $\lambda$ real. Eigenvalue lower-bound conjecture. There exists a constant $\delta>0$ depending only on $K$, $C$, $\operatorname{diam}(M)$, $\operatorname{inj}(M)$, and $n$ such that $\lambda>\delta$.

This conjecture seeks to replace the two-sided Ricci-curvature control used earlier with a lower Ricci bound alone, while retaining dependence on the injectivity radius. The authors motivate it as a possible extension of their eigenvalue estimates beyond the setting where a drift ansatz provides the needed regularity; its resolution is not given here.