Eigenvalue convergence for kernels under total-variation convergence

Let $(X,d,\mu)$ be a bounded metric measure space with kernel $K$ and associated operator $T_K\colon L^2(X,\mu)\to L^2(X,\mu)$. Let $(X,d,\mu_n)$ have analogous kernel $K_n$ and operator $T_{K_n}\colon L^2(X,\mu_n)\to L^2(X,\mu_n)$, and let $\lambda(T_{K_n})$ and $\lambda(T_K)$ denote their ordered spectra. Suppose that $\mu_n\to\mu$ in total variation. Eigenvalue-convergence conjecture. The ordered spectrum of $T_{K_n}$ converges to that of $T_K$ in $\a0ell^2$-distance:

$$\ell^2(\lambda(T_{K_n}),\lambda(T_K))\to 0.$$

This conjecture removes the eigenvalue-convergence hypothesis from the preceding proposition, which concerns convergence of eigenfunctions for operators associated with measures converging in total variation. The source mentions that ideas from Koltchinskii (2000) may be relevant, but the claim remains open here.