Spectrum occupies pseudospectrum for random matrices with diagonal deformation and variance profile

We consider $n\times n$ non-Hermitian random matrices with independent entries and a variance profile, as well as an additive deterministic diagonal deformation. We show that their empirical eigenvalue distribution converges to a limiting density as $n$ tends to infinity and that the support of this density in the complex plane exactly coincides with the $\epsilon$-pseudospectrum in the consecutive limits $n\to \mathrm{\infty }$ and $\epsilon \to 0$. The limiting spectral measure is identified as the Brown measure of a deformed operator-valued circular element with the help of [arXiv:2409.15405].