Poincaré-type stability conjecture for weighted Einstein manifolds

Let $(M^n,g,v,m,0)$ be a closed weighted Einstein manifold with $P_\phi^m=\lambda g>0$ and scale $\kappa>0$. Let $I_1,I_2\colon C^\infty(M)\to\mathbb{R}$ be the functionals defined by the weighted Einstein eigenvalue formulas. Then

$$\inf\left\{I_j[\psi]\mathrel{}\middle|\mathrel{}\int_M\psi\,d\nu=1\right\}>0$$

for $j\in\{1,2\}$. In particular, the Hessian

$$D^2\mathcal{Y}_k\colon T_{(g,v)}\mathfrak{C}\to\mathbb{R}$$

is positive definite. Poincaré-type stability conjecture. Every closed weighted Einstein manifold satisfying $P_\phi^m=\lambda g>0$ and having scale $\kappa>0$ satisfies these positivity inequalities, and consequently is stable for the corresponding $\mathcal{Y}_k$-functional. This is expected by analogy with stability results for quasi-Einstein manifolds; the conjecture concerns the cases in which the weighted $\sigma_k$-curvatures are variational, and no resolution is supplied here.