The real-part spectral-symbol conjecture for matrix sequences

Let $\{X_n\}_n$ be a matrix sequence with spectral symbol $f$, written

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

Assume that the imaginary parts satisfy

$$\|\operatorname{Im}(X_n)\|_1=o(n).$$

Real-part spectral-symbol conjecture. Is it true that

$$\{\operatorname{Re}(X_n)\}_n\sim_\lambda f?$$

The question asks whether a matrix sequence whose imaginary parts are small in trace norm has the same spectral symbol as its Hermitian real part. The source identifies this as an opposite problem related to the preceding perturbation conjecture, and the supplied parser status gives no evidence of resolution.