Improved homogenization bound for Steklov eigenvalues with rapidly oscillating weights

Let $\lambda_{k,\varepsilon}$ and $\lambda_{k,0}$ denote the $k$-th Steklov eigenvalues, respectively, for the rapidly oscillating boundary weight $\rho_\varepsilon$ and its homogenized limit $\rho_0$, and let $C(\Omega)$ be the constant appearing in Theorem 2. The conjectured estimate concerns the difference between these eigenvalues. Improved Steklov homogenization conjecture. The bound should be improved at least to

$$|\lambda_{k,\varepsilon}-\lambda_{k,0}|\leq C(\Omega)\sqrt{\varepsilon}\,k^{\frac{2}{n-1}}.$$

This conjecture is motivated by the sharper bounds obtained for Dirichlet and Neumann eigenvalues. The supplied text does not state whether this improvement has been proved or disproved.