Steklov critical metrics versus critical metric–density pairs

Let $\Omega$ be a compact manifold of dimension $d + 1 \ge 2$ with boundary $\Sigma$. For a Riemannian metric $g$ on $\Omega$ and a positive smooth density $\rho$ on $\Sigma$, let $\sigma_k(\Omega, g, \rho)$ be the $k$-th eigenvalue of the weighted Steklov problem $\Delta_g u = 0$ in $\Omega$, $\partial_\nu u = \sigma \rho u$ on $\Sigma$, and let $$\bar\sigma_k(\Omega, g, \rho) = \sigma_k(\Omega, g, \rho)\, |\Omega|_g^{(1-d)/(d+1)}\, \|\rho\|_{L^1(\Sigma, g)}, \qquad \bar\sigma_k(\Omega, g) = \bar\sigma_k(\Omega, g, 1),$$ the normalisation of Karpukhin and Métras. A pair $(g, \rho)$ is $\bar\sigma_k$-critical if for every smooth deformation $(g_t, \rho_t)$ of it either $\bar\sigma_k(g_t, \rho_t) \le \bar\sigma_k(g, \rho) + o(t)$ or $\bar\sigma_k(g_t, \rho_t) \ge \bar\sigma_k(g, \rho) + o(t)$, and $\bar\sigma_k$-conformally critical if this holds for the deformations with $g_t \in [g]$; a metric $g$ is ($\bar\sigma_k$-conformally) critical if the same holds for $\bar\sigma_k(g_t)$ under deformations of the metric alone. Karpukhin and Métras characterised critical pairs by free boundary minimal immersions into balls, and conformally critical pairs by free boundary $(d+1)$-harmonic maps. If $(g, 1)$ is a critical (respectively conformally critical) pair, then $g$ is a critical (respectively conformally critical) metric. The converse holds when $\sigma_{k-1}(\Omega, g) < \sigma_k(\Omega, g)$ or $\sigma_k(\Omega, g) < \sigma_{k+1}(\Omega, g)$.

Problem. If $g$ is a $\bar\sigma_k$-critical metric, is $(g, 1)$ necessarily a $\bar\sigma_k$-critical pair? If $g$ is a $\bar\sigma_k$-conformally critical metric, is $(g, 1)$ necessarily a $\bar\sigma_k$-conformally critical pair?