Uniform comparison of all Steklov eigenvalues with those of a discretisation

For a finite connected graph $\Gamma$ with a set $B$ of boundary vertices, let $$\sigma_k(\Gamma) = \min_{F} \max_{0 \ne u \in F} \frac{\sum_{i \sim j} (u(i) - u(j))^2}{\sum_{i \in B} u(i)^2},$$ the minimum running over the $(k+1)$-dimensional spaces $F$ of functions on the vertices; there are $|B|$ such eigenvalues. For $\kappa \ge 0$ and $r_0 \in (0, 1)$, let $\mathcal M(d, \kappa, r_0)$ be the class of compact Riemannian manifolds $\Omega$ of dimension $d + 1$ with nonempty boundary $\Sigma$ such that a neighbourhood of $\Sigma$ is isometric to the cylinder $[0, 1] \times \Sigma$, $\operatorname{Ric}_\Omega \ge -d\kappa$, $\operatorname{Ric}_\Sigma \ge -(d-1)\kappa$, the injectivity radius of $\Omega$ exceeds $r_0$ at points at distance more than $1$ from $\Sigma$, and that of $\Sigma$ exceeds $r_0$ everywhere. Colbois, Girouard and Raveendran associated with each $\Omega \in \mathcal M(d, \kappa, r_0)$ and each mesh $\varepsilon \in (0, r_0/4)$ discretisations $\Gamma$, graphs with boundary built from maximal $\varepsilon$-separated sets, and proved that there are constants $A_1, A_2 > 0$ depending only on $d, \kappa, r_0, \varepsilon$ with $$\frac{A_1}{k} \le \frac{\sigma_k(\Omega)}{\sigma_k(\Gamma)} \le A_2 \qquad \text{for } 1 \le k \le |B| - 1,$$ where $B$ is the boundary of $\Gamma$, and that the lower bound can be taken independent of $k$ for those $k$ with $\sigma_k(\Omega) \le C/k$, where $C$ depends only on $d, \kappa, \varepsilon$. The analogous comparison for the Laplacian holds with constants independent of $k$.

Problem. Are there constants $A_1, A_2 > 0$, depending only on $d, \kappa, r_0, \varepsilon$, such that $$A_1 \le \frac{\sigma_k(\Omega)}{\sigma_k(\Gamma)} \le A_2$$ for every $\Omega \in \mathcal M(d, \kappa, r_0)$, every such discretisation $\Gamma$ of $\Omega$ and every $1 \le k \le |B| - 1$?