The small trace-norm perturbation conjecture for Hermitian matrix sequences

Let $X_n$ be a Hermitian matrix of size $n$, and let $\{X_n\}_n\sim_\lambda f$. Let $Y_n$ be a matrix sequence satisfying

$$\|Y_n\|_1=o(n).$$

Small trace-norm perturbation conjecture. Then

$$\{X_n+Y_n\}_n\sim_\lambda f.$$

This would strengthen known perturbation results for Hermitian matrix sequences, which require either a smaller trace-norm perturbation or additional uniform spectral-norm bounds. Its status is not resolved in the supplied source.