Effective maximum-principle conjecture for eigenfunctions

Let $M$ be a compact Riemannian manifold with smooth boundary, nonnegative Ricci curvature, bounded second fundamental form of the boundary, and diameter $1$. Let $f$ be a function on $M$ satisfying

$$\Delta f=\lambda f.$$

Suppose that for sufficiently small $\varepsilon>0$,

$$\|f\|_{L^\infty(\partial M)}\leq\varepsilon,$$

and that $D\subset M$ has large measure and satisfies

$$\|\nabla f\|_{L^\infty(D)}\leq\varepsilon.$$

Effective maximum-principle conjecture. Then

$$\|f\|_{L^\infty(M)}\leq \frac{\Psi(\varepsilon\mid n,\lambda)}{\Psi_1(\operatorname{meas}(D)\mid n,\lambda)}.$$

Here $\Psi$ and $\Psi_1$ are the functions introduced in the source. This conjecture seeks a quantitative form of the maximum principle: small boundary values and small gradient on a sufficiently large subset should force a uniformly small eigenfunction. It is used to explain the remaining difficulty in proving the corresponding essential-spectrum result.