Stability conjecture for higher Laplace eigenvalues on the sphere

Let $g_0$ be a round metric of curvature one on $\mathbb{S}^2$. Given $k\geq 2$, let $\mu$ be an admissible measure such that $\lambda_k(\mathbb{S}^2,\mu)=2$. Then there exists a conformal automorphism $\Phi\colon\mathbb{S}^2\to\mathbb{S}^2$ and $m\leq k-1$ distinct points $p_1,\dots,p_m\in\mathbb{S}^2$ such that

$$8\pi k-\bar\lambda_k(\mathbb{S}^2,\mu)\geq c\left\|dv_{g_0}+\sum_{i=1}^m4\pi\alpha_i\delta_{p_i}-\Phi_*\mu\right\|^2_{(C^1(\mathbb{S}^2))^*}$$

for some constant $c>0$ and positive integers $\alpha_i$ satisfying $\sum_{i=1}^m\alpha_i=k-1$. Higher-eigenvalue stability conjecture. The deficit from the sharp upper bound $8\pi k$ controls the squared dual $C^1$-distance from $\Phi_*\mu$ to the measure consisting of the round area measure together with at most $k-1$ weighted point masses, whose positive integer weights sum to $k-1$. The proposed extension is motivated by the known upper bound and the limiting configuration of $k$ identical round spheres, but the argument is currently available only for $k=1,2$.