Wang–Xia's conjecture on the first Steklov eigenvalue

Let $(M^n,g)$, with $ngeq 2$, be a connected compact smooth Riemannian manifold with boundary. Assume that $\operatorname{Ric}_g\geq 0$ and that the principal curvatures of the boundary $\partial M$ are bounded below by a constant $c>0$. Denote by $\lambda_1=\lambda_1(\partial M)$ the first nonzero eigenvalue of the Laplacian of $\partial M$. Let $\xi_1$ be the first nonzero Steklov eigenvalue.

Wang–Xia's conjecture. The eigenvalue $\xi_1$ satisfies

$$\xi_1\geq \frac{n+2}{n-1}c\lambda_1,$$

with equality only for the Euclidean ball of radius $1/c$.

This conjecture was stated by Qiaoling Wang and Changyu Xia and is verified in the paper for warped product manifolds when $n=3$ and $m=1$. Its status in the generality stated above is not resolved by the supplied source.