Quantitative stability conjecture for optimal potentials on general domains

Let $\Omega$ be a domain, let $V_\Omega$ be a minimizer of the variational problem, and let $u_\Omega$ be its associated eigenfunction. Write

$$V_\Omega=\chi_{E_\Omega}=\chi_{\{u_\Omega\geqslant\mu_\Omega\}}$$

for some $\mu_\Omega\in\mathbb R$. Assume that the minimizer is regular, in the sense that

$$\frac{\partial u_\Omega}{\partial\nu}\leqslant -C<0$$

on $\partial E_\Omega$, and that $E_\Omega$ is a non-degenerate shape minimizer: for every admissible variation $\Phi\in\mathcal X_1(E_\Omega)$, if $L_\tau$ is the associated Lagrangian, then

$$L_\tau”(E_\Omega)[\Phi,\Phi]>0.$$

Quantitative stability conjecture. There exists $\eta>0$ such that, for every $V\in\mathcal M(\Omega)$,

$$\|V-V_\Omega\|_{L^1(\Omega)}\leqslant\eta\ \Longrightarrow\ \lambda(V)-\lambda(V^*)\geqslant C\|V-V^*\|_{L^1(\Omega)}^2.$$

This conjecture proposes extension of the quantitative inequality from the ball to general domains under regularity and non-degeneracy assumptions on the minimizing shape. The notation $V^*$ in the final inequality appears inconsistent with the preceding general-domain notation, where the natural reference potential is $V_\Omega$; this should be checked against the source.