The weighted orthogonality Poincaré inequality conjecture

Let $(M^n,g)$ be a compact Riemannian manifold with $\partial M=\emptyset$. Let $\lambda_1$ be the first eigenvalue of the Laplace operator $\Delta$ on $(M^n,g)$, and let $\mathscr{U}_1$ be the first eigenspace of $\Delta$. Let $f:M\rightarrow\mathbb{R}^+$ be a fixed smooth function satisfying $f\perp\mathscr{U}_1$. For a smooth function $\phi:M\rightarrow\mathbb{R}$ satisfying

$$\int_M f\phi\,d\mu=0,$$

Weighted orthogonality Poincaré conjecture. One has

$$\lambda_1\int_M\phi^2\,d\mu\leq\int_M|\nabla\phi|^2\,d\mu.$$

This is proposed as a higher-dimensional analogue of the paper's Poincaré-type inequality on $S^1$, and parallels the usual Poincaré inequality governed by the first Laplace eigenvalue. The source presents it as a proposed conjecture; no resolution is supplied in the given text.